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Elements of Abstract Harmonic Analysis ACADEMIC PAPERBACKS’ BIOLOGY Edited by ALVIN NASON Design and Function at the Threshold of Life The Viruses Heinz Fraenkel-Conrat Time, Cells and Aging Bernard L Strehler Isotopes in Biology George Wolf Life Its Nature, Origin, and Development A I Oparin MATHEMATICS Edited by W MAGNUS and A SHENtTZER Finite Permutation Groups Helmut Wielandt Introduction to p Adic Numbers and Valuation Theory Georce Bachman Quadratic Forms and Matrices N V Yefimov Elements of Abstract Harmonic Analysis George Bachman Noneuclidean Geometry Herbert Meschkowski PHYSICS Edited by D ALLAN BROMLEY Elementary Dynamics of Particles H W Darkness Elementary Plane Rigid Dynamics H W Harkness Crystals Their Role in Nature and in Science Charles Bunn Potential Barriers in Semiconductors B R Gossick Mftssbauer Effect Principles and Applications Gunther K Wertkeim * Most of these volumes are also available in a cloth bound edition Elements of Abstract Harmonic Analysis By George Bachman POLYTECHNIC INSTITUTE OF BROOKLYN DEPARTMENT OF MATHEMATICS BROOKLYN, NEW YORK with the assistance of Lawrence Narici POLYTECHNIC INSTITUTE OF BROOKLYN DEPARTMENT OF MATHEMATICS BROOKLYN, NEW YORK ACADEMIC PRESS • New York and London Copyright © 1964 by Academi ALL BIGHTS RESERVED *S«SB W#St »« 4 " ANT T0RM BY PHOTOSTAT MICROFILM 08 ~ t OTHX* *00X3 WTTHOtrr WHITTEN PERMISSION PROM THE PUBLISH** 3 ACADEMIC PRESS INC UI Fifth Avenue New York New York NXXtf United Kingdom Edition published by ACADEMIC PRESS INC (LONDON) LTP Berkeley Square House London W l Library or Congress Catalog Car& Num^* b 64-21663 PRTNTKO IS THU raniB STATES AMERICA Preface Abstract Harmonic Analysis is an active branch of modern analysis which is increasing in importance as a standard course for the beginning graduate student. Concepts like Banach algebras, Haar measure, locally compact Abelian groups, etc., appear in many current research papers. This book is intended to enable the student to approach the original literature more quickly by informing him of these concepts and the basic theorems involving them. In order to give a reasonably complete and self-contained introduction to the subject, most of the proofs have been presented in great detail thereby making the development understandable to a very wide audience. Exercises have been supplied at the end of each chapter. Some of these are meant to extend the theory slightly while others should serve to test the reader’s understanding of the material presented. The first chapter and part of the second give a brief review of classical Fourier analysis and present concepts which will subsequently be generalized to a more abstract framework. The presentation of this material is not meant to be detailed but is given mainly to motivate the generaliza- tions obtained later in the book. The next five chapters pre- sent an introduction to commutative Banach algebras, gen- eral topological spaces, and topological groups. We hope that Chapters 2-6 might serve as an adequate introduction for those students primarily interested in the theory of com- mutative Banach algebras as well as serving as needed pre- requisite material for the abstract harmonic analysis. The remaining chapters contain some of the -measure theoretic background, including the Haar integral, and an extension of the concepts of the first two chapters to Fourier analysis on locally compact topological abelian groups. In an attempt to make the book as self-contained and as introductory as possible, it was felt advisable to start from scratch with many concepts — in particular with general Preface topological spaces However within the space limitations it was not possible to do this with certain other background material— notably some measure theory and a few facts from functional analysis Nevertheless the material needed from these areas has all been listed in various appendices to the chapters to which they are most relevant There are now a number of more advanced books on abstract harmonic analysis which go deeper into the subject We cite in particular the references to Rudin Loomis and the recent book by Hewitt and Ross Our treatment of the latter part of Chapter 12 follows to some extent the initial chapter of the book by Rudin which would be an excellent continuation for the reader who wishes to pursue these matters further The present book is based on a one semester course m abstract harmonic analysis given at the Polytechnic In stitute of Brooklyn during the summer of 1963 for which lecture notes were written by Lawrence Narici A few re visions and expansions have been made I would like to express my sincere gratitude to Mr Nanci for bis effort in writing the notes improving many of the proofs and for editing the entire manuscript I would like finally to express also my appreciation to Melvin Maron for his help m the preparation of the manuscript George Bachman Contents PREFACE V SYMBOLS USED IN TEXT X Chapter 1 The Fourier Transform on the Real Line for Functions in Li 1 Introduction 1 Notation 1 The Fourier Transform 2 Recovery 4 Relation between the Norms of the Fourier Transform and the Function 10 Appendix to Chapter 1 15 Exercises 17 References IS Chapter 2 The Fourier Transform on the Real Line for Functions in l 2 19 Inversion in hi 21 Normed and Banach Algebras 25 Analytic Properties of Functions from C into Banach Algebras 29 Exercise 33 References 33 Chapter 3 Regular Points and Spectrum 34 Compactness of the Spectrum 38 Introduction to the Gel’fand Theory of Commutative Banach Algebras 48 The Quotient Algebra 50 Exercises 53 References 54 Chapter 4 More on the Gel'fand Theory and an Intro- duction to Point Set Topology 55 Topology 60 A Topological Space 61 vii Contents Examples of Topological Spaces 61 Further Topological Notions 62 The Neighborhood Approach 66 Exercises 71 References 72 Chapter 5 Further Topological Notion* 73 Bases, Fundamental Systems of Neighborhoods, and Subbases 73 The Relative Topology and Product Spaces 78 Separation Axioms and Compactness 79 The Tychonoff Theorem and Locally Compact Spaces 84 A Neighborhood Topology for the Set of Maximal Ideals over a Banach Algebra 87 Exercises 89 References 90 Chapter 6 Compactness of the Space of Maximal Ideals over a Banach Algebra; an Introduction to Topological Groups and Star Algebras 91 Star Algebras 97 Topological Groups 98 Exercises 106 References 106 Chapter 7 The Quotient Group of a Topological Crovp and Some Further Topological Notions 107 Locally Compact Topological Groups 107 Subgroups and Quotient Groups 109 Directed Sets and Generalized Sequences 116 Further Topological Notions 117 Exercises 123 References 124 Chapter 8 Right Haar Measures and the Haar Covering Function 125 Notation and Some Measure Theoretic Results 125 The Haar Covering Function 129 Summary of Theorems in Chapter 8 147 Exercises 149 References 149 Contents ix Chapter 9 The Existence of a Right invariant Haar Integral over any Locally Compact Topolog- ical Group 150 The Daniell Extension Approach 158 A Measure Theoretic Approach 160 Appendix to Chapter 9 163 Exercises 164 References 165 Chapter 10 The Daniell Extension from a Topological Point of View, Some General Results from Measure Theory, and Group Algebras 166 Extending the Integral 166 Uniqueness of the Integral 169 Examples of Haar Measures 172 Product Measures 176 Exercises 186 References 187 Chapter 1 1 Characters and the Dual Group of a Locally Compact, Abelian, Topological Group 188 Characters and the Dual Group 192 Examples of Characters 200 Exercises 206 References 206 Chapter 12 Generalization of the Fourier Transform to Li(G) and L 2 (G) 20/ The Fourier Transform on Li(G) 207 Complex Measures 211 The Fourier-Stieltjes Transform 219 Positive Definite Functions 220 The Fourier Transform on Lj(G) 235 Exercises 243 Appendix to Chapter 12 244 References 250 BIBLIOGRAPHY 251 INDEX 253 Symbols Used in Text \x i pi / X —*Y x-*y CE 0 c The set of all x with property p The function / mapping the set X into the set Y where x £ X is mapped into y 6 Y Further, if E C X,f(E) will denote the image set of E under /, ie, /(£) =* j/{x) J x € E) If M C will denote those x € X such that /(x) € 21/ The notation / |b, E C X, will denote the restriction of / to E The complement of the set E The null set The complex numbers In the list below the number immediately following the symbol will denote the page on which the symbol is defined The symbols are bsted according to the order m which they appear in the text R, 1 Ep, 1 11/ Ik 2, J)/ IJ., 180, 11/ II., 196 / ' g, 6, 180 /, 2 L(X, X), 26 Li(Z), 28 |M ||, A€L(X,X), 26 C£a, bj, 26 W, 26 Z, 28 <r(x), 37 Rx, 41 r.(x), 42 M, 52 x(M), M €. M, 51 F(&), 57 6, 61 E, 62 E°, 71 V{x), 66 V{M,x h x t , •••,*., «), 87 C*, 99 R*, 99 ViV t - |xy|z€ V,,y€ V,}, 100 [/-* = ix~'\x€ V), 100 G/H, 110 lim /(s), 116 s X*, 118 C, 125 ft 126 C«(G), 129 C 0 + (G), 129 k E , 129 (/:*), 130 I, if), 136 I if), 157 /**, 161 E z , E y , 177 f x , U 177, 188 U{G), 180 Symbols Used in Text xt f*(x), 188 R(G), 193 x, 193 fix), 193 N( X o,E, e), 204 L 2 (G), 220 B(X), B(Y), 212 fi * v, 214 Nix 0 ,E,e), 222 CHAPTER 7 The Fourier Transform on the Real Line for Functions in Li Introduction This section will be devoted to some Fourier analysis on the real line. The notion of convolution will be introduced and some relationships between functions and their trans- forms will be derived. Some ideas from real variables (e.g. passage to the limit under the Lebesgue integral and inter- changing the order of integration in iterated integrals) will be heavily relied upon and a summary of certain theorems that will be used extensively in this section are included in the appendix at the end of this chapter. Notation Throughout, R will denote the real axis, ( — «> , a> ) , and / will denote a measurable function on R (f may he real or complex valued) . We will denote by L p the set of all meas- urable functions on R with the property! | f(x) \ F dx < co , 1 < p < CO. Also, since we will so frequently be integrating from minus infinity to plus infinity, we will denote f by simply f ^ — co J where all integrations are taken in the Lebesgue sense. t One often says that / is pth power summable. Element* of Abstract Harmonic Analysis Noting without proof that the space L p is a linear space we can now define the norm of f with respect to p, / j) p , where / 6 L p as follows 11/11, = (/ 1 m I ’**)'" It is simple to verify that the following assertions are valid 1 I! /II* ^ 0 and |( /(ip = 0 if and only if f ~ Ot 2 II VII, “ Ul ll/ll, Where l is a real or complex number Note that this immediately implies (1/ |j p = 11 -/ 11 * 3 11/ + 9 lie < ll/ll, + II 9 ll» To verify this all one needs is Minkowski’s inequality for integrals which is listed m the appendix to this chapter (p 15) Thus it is seen that L„ is a normed linear space One can go one step further however although it will not be proven here it turns out that L, is actually a complete normed linear space or Banach space or that any Cauchy sequence (in the norm with respect to p) of functions in L p comerges (in the norm with respect to p) to a function that is pth power summable The Fourier Tronsform In this section we consider the Fourier transform of a function / in Li and note certain properties of the Fourier transform Let / 6 Lj and consider as the Founer transform off m = /,■"/(() a t We mil say that two functions / and q are equivalent, denoted by / ~ 0, if / = g almost everywhere Thus I P is actually the set of all equivalence classes of functions tinder this equivalence relation If we ’utA, tills, to 1) -wttoWi TtprtiiTA. ■& paeudw.cm because V. would be possible for J)/)|> to equal zero even though/ was not zero 1. Fourier Transform for Functions in Li 3 We first note that f(x) exists for, since / 6 L h I /(*) I < / \m\dt < oo. Further, since / 1/(0 I* = II / IK we can say \f(x) I < 11/ 111 for any x ; or that the 1-norm of / is an upper bound for the Fourier transform of /. Since it is an upper bound it is cer- tainly greater than or equal to the least upper bound or sup | J{x) | < || /||i. xzR It will now be shown that the Fourier transform of / is a continuous function of x. Consider the difference fix + h„) - f{x) = J e' xi (e ih "‘ - l)/(<) dt where h„ € R: \f(x + h„) - f(x) | < J | expiihj) - 1 | | fit) | dt. Since this is true for any h„ it is true in the limit as h n ap- proaches zero, or lim (/(s + h„) — fix) | < lim f | e ihnt — 1 | | /(<) | dt. kn~+f) hn~*0 Our wish now is to take the limit operation under the inte- gral sign, and Lebesgue's dominated convergence theorem will allow this manipulation (see appendix to Chapter 1) for certainly the integrand in the last expression is dominated dementi of Abstract Harmonic Analysis by the summable function 2 |/(f) | Taking the limit inside yields the desired result, namely, that lim /(* + K) - /(x) or that the Fourier transform / of a function / m I«i is a con- tinuous function The following fact can also be demonstrated about /(x) and is usually referred to as the Riemann-Lebesgue lemma lim /( jc) = 0 (1) We note in passing that there are continuous functions, F(i), satisfying (1) but such that no /(<) can be found that satisfies F(i) - j «“'/(<) dt This bnngs us to our next problem knowing /(x), how can the function that it came from,/(t), be found again? Recovery In elementary treatments one often sees the following inversion formula /(<) “ ~ /' dz It will now be demonstrated, by a counterexample, that the above formula is not true in general Example Consider the function (e~‘, t > 0 m = i ip, t < 0 /(x) - f '%«-»■<« - — J o XX — 1 1 . Fourier Transform for Functions in L j 5 It is now clearly impossible to recover /(f) using the formula for / | | lx - /-^=_ . Since the last integrand behaves like 1/a:, for large x the inte- gral will become infinite as log x and, therefore, we cannot recover /(f) using that formula. (Note that a Lebesgue integral must be absolutely convergent in order to converge.) Before proceeding further, two results from real variables will be needed: Definition, t is said to be a Lebesgue point of the function f if 1 f t+h iim i / I /(«) ~ f(x) 1 dx = 0. A-0 “ J t Theorem. If / € L lt then almost all of its points are Lebesgue points. Theorem. Every point of continuity of a function is a Lebesgue point. The following two theorems on inversion are lengthy and somewhat intricate and the reader is referred to Goldberg [ ’2] for the proofs. Theorem 1 . Let /, / 6 Li and suppose / is continuous at t, then /(f) = — f er** l f(x) dx. Theorem 2. Let f € Li and let f be a Lebesgue point for the function, /, then /(f) = lim — f (l - — V fa '/(aO dx. (Note that this limiting process is analogous to ( C , 1) sum- mability criterion for infinite series.) Elements of Abstract Hormonic Anolysu Corollary 1. Suppose / 6 Za and f(x) — 0, for all x, then f(t) = 0,ae Corollary 2. Suppose/i,/i £ L\ If /i = then/i(t) = /i(t) ae This result follows immediately from Corollary 1, for we have /\ u-u-h -/. -o therefore /ir/i B 0. a e Convolution Let f g € Li and consider the function fc(x) - f}(x - 1)9(1) <fl = (/-sKi) and called the conuolufton o/ / tattfc p It is now contended that k(z) exists for almost all x and that k(z) is summable Proof It is easily shown that ff{x - t)dx = Jf{ x) dx (2) by simply making a change of variable Now consider fdtf |/(x - 1)9(1) I dx - / | j(l) | dlf |/<x - I) | dx -Il9ll.il/ll. < ” by usmg (2) Now, by the Tonelli-IIobson theorem (see appendix to Chapter 1), it follows that jjf(x - t)g(t) dtdx is absolutely convergent By the Fubim theorem it follows that h(x) exists a e and is integrable It will now be shown that the operation of convolution is a commutative one Theorem / » g = g */for /, g £ Z* 1. Fourier Transform for Functions in Li 7 Proof. if *g) (x) = Jf(x ~ t)g(t) dt. Let u = x — t. Then (f * (j) ( x ) = / f(u)g(x - u)(-du ) = (<?*/) (s). ~ CO It also follows that the operation of convolution is associa- tive or f*(g*h) = U*g) *h where /, g, h 6 Lj. The proof of this result, although straightforward, is rather messy and will be omitted. Theorem. Let f, g (• la. Then \\f*g\\i < 11/ Hill g Ili- Proof. II/* S' Hi = Jdx\ Jf(x - t)g(t) dt | < fdxj |/(x - t)g(t) | dt. (3) We now note that Jdtj | f(x - t)g(t ) | dx = J | g{t) | dtj | f(x - t) | dx = lUllill/lli < 00 ( 4 ) or that (3) converges absolutely. By the Tonelli-Hobson theorem,! Jdxj | f{x — t )g{t) | dt converges absolutely and, by the same theorem must also t See appendix to Chapter 1. Element] of Abstract Harmonic Analysis converge absolutely to the value jdtf |/(l - t)j(i) \it - l!/|!i(UH, which yields the desired result We can summarize the above results by noting that la with respect to the operations of addition and convolution forms a Banach algebra (see Chapter 2, Example 6) The next theorem is one of the mam reasons for interest in convolution Theorem. Let / g € Lj, then /\ , / • » ~ a Proof /\ </ • 5) (*) Since fdsf |e“'/(t - =* dl «=> je al dtjf(t - s)g{s) ds (5) s)g(s)\dt=* jdej \f(t ~ s)g(s) \ dt = / \9{*)\dsj « II 9 Hill /Hi < • then, by the same reasonuig as m the previous theorem (use of the TonelU-Hobson theorem), the order of integration may be interchanged in (5) Wnte e u ‘ = 1. Fourier Transform for Functions in Li 9 substitute this in (5), and interchange the order of integra- tion. Now (/* 9 ) 0*0 = / ff (•)«** ds Jf(t - s)e' z "-‘ ) dt= gf = fg. QED We noted that Li(+, *) was a Banach algebra. We will now show that it is not an algebra with identity for suppose there were an element, e € Lj, such that f *e — J for every / 6 L 2 . Then certainly e * e = e. By the previous theorem, then /\ e * e = — e. Therefore t — 0, 1 are the only possibilities. By continuity of 8, e must equal either zero or one; it cannot jump! But, since we require in addition lim e(x) = 0, X-*CO we must choose i(x) = 0. This implies that e(f) = 0, a.e. and the only way that this could be an identity would be if f(t) = 0, a.e., for all / 6 la, which is ridiculous. Even though there can be no identity there is what is called an approximate identity, there exists a sequence of functions, {e„} in Li such that lim„^„ || e n *f — / ||i = O.f t See Goldberg [2], ?0 Elements of Abstract Harmonic Analysis Relation between the Norms of the Fourier Transform and the Function Before proceeding to the main result we need the following Lemmo 1 Let a, b 6 B, b > 0 Then /«"'exp(-W’) it - (0 ' exp (-f) The proof will only be outlined Consider J exp (— 2 *) dz where z is a complex variable and F ts the contour shown below Splitting up the integral into and taking the limit as k — * «> yields the desired result We must also establish the following limit before proceeding to the theorem Lemma 2 Let/ € Li n L t , then lim j exp (—**/«) )/(*) ]*di = 2ir ]) / ))| T . Fourier Transform for Functions in ii 11 Proof. Certainly I /(*) I 2 = f(x)](x) = J f(t)e ixl dtj f(s)er ix ‘ ds. Multiplying both sides by exp ( — x 2 /n) , where n is an integer, and integrating with respect to x, we have J exp(—x 2 /n) ) f(x) | 2 da: = J exp (—x 2 /n)dxjf(t)e' x, dtjf(s)e~ ix ’ds. By the exact same reasoning used before we can interchange the order of integration to get y"exp( — x 2 /n) | f(x) | 2 dx = ff(s) ds jf(t) dt Je ix(t ~’ } exp(— x 2 /n) dx. Using the result of Lemma 1 with a — t — s, b = 1/n, x = t we can evaluate the last integral on the right as y^ e ivr(i-j) exp(— x 2 /n) dx — (mi) 112 exp( — n(t — s) 2 /4) or J exp (— x 2 /n) [ f(x) \ 2 dx = (xn)» 2 J/( S ) dsjf(t) exp (-n(i - s) 2 /4) dt. (*) 12 Element* of Absfroct Harmonic Analysis Interchanging the order of integration [see Eq («) below for the justification]] we get j exp (-!*/«) l/(x) \ a dx «= (t n)* n j exp (— n£*/4) dtjf(t + s)/(s) ds after first replacing / by t + s in (^) Let us denote by g(t) 9(0 - //« + *)/(<) d* or j exp(— z J /n) !/(*) |* dr = (s exp (— nt 1 / 4 ) * Replace / by 2n~ w / This gives for the preceding integral 2r'” J g(2n~ ll H) exp( -t 1 ) dt We now claim that g(t) is continuous at the origin Hence we must show lim | g(t ) - g( 0 ) | =0 I 9(0 - 9(0) |> - | //(»)(/(< + «) -/(«)) *1* < / |/<5) |’<b- / |/(l + *) -/W I'*. and since hm j j f(t + s) - /(«) | l ds = 0 1. Fourier Transform for Functions in L i 13 (see exercise 4) we therefore have lim | g(l) - g(0) | = 0. t -+ 0 Thus g(t) is continuous at the origin. Now I g(t) I = J ff(t + s)f(s) ds < J \f(t + s)f(s) | ds < (/ !/ ( t + s) |* *)”*(/ l/(s) \ 2 dsJ 2 by the Cauchy-Schwarz inequality. Thus MO I < 11/11*11/11* = 11/115 <•) for any t. Now we can say for any n and, in particular, in the limit as n — * oo , lim J exp (— x 2 /n) | f(x) | 2 dx n-*a 3 = lim 27 r ll2 J exp( — t 2 )g(2n~ ll H) dt. n-*a> Now since 1 1 1 11 exp ( — t 2 ) is summable and dominates the last integrand we can say lim J exp(— x 2 /n) \ f(x) | 2 dx = 2rg(0) n-+ co J i = 2x |1 / 111- We can now proceed to the main result. Theorem. Let/ £ Li fl L 2 , then / € Lia,n&\\} \\\ = 2ir\\f\\i. 14 Elements of Abstract Harmonic Analysis Proof j \j{ x) j *dx = f limexp(-z*/n) ] /(ic) J*dx since lmi ll -. a ,evp(— x*/n) = 1 Now since the sequence of functions (exp(— x*/n) | f(x) [*} is nonnegative and also monotone increasing so that supjj exp(-x*/n) |/(x) |* rfxj = Iim Jexp(-x*/ n ) |/(x) (Vx ne can apply Fatou’s Jemma t and say J 1 /(*) |*dx < lun J exp(— x*/n) 1 /(*) l 5 dx “ 2*- U/lll by Lemma 2 This proves that / € Lj, or that ( / f 1 is sum- mable But since )/|* is sumniable, then 1*1/}* dominates exp(— x*/n) l/(x) |* Therefore lun j e\p( — x*/») I /(x) 1* dx => j lim |/(x) )*exp( — x*/n) dx or /|/(*)|><fc-|l/l|S = 2'll/ll» QED Definition. Consider a function /(t) Define [m, ui<« /-(*) - [0, t > n Theorem. Let / € L*, then/, € L\ n Li and /, € Li for all li n t positive integers n and the sequence /, » a function m Lt f i e m the mean square sense) t See appendix to Chapter 1 1 . Fourier Transform for Functions in Li 15 Proof. / ifn(t) | dt = f \f(t) | dt — R < (/" 1/(0 r-dtj /2 (fi.dtj' 2 < l|/||.(2n)‘« or /„ 6 Li. Clearly, /„ £ Z, 2 . Therefore /„ 6 Li n Z/ 2 . By the previous theorem /n € £2- We will now show that the sequence { /„} is a Cauchy sequence; i.e. lim || /„ - f n || 2 = 0. n,m-* co Since /„ — f m 6 L t D L 2 , we can say II A - ||l = 2ir ||/„ — f m ||I. But (assume n > m) [ 1 1 1 > n fn — fm ~ 0 if 1 1 1 1 < m so ll/n - fm Hi = [ m \f(t)\ 2 dt + f\m | 2 dl, but since / 6 L 2 , each of the above terms must go the zero as n, m — » <». Thus { /„} is a Cauchy sequence in L 2 . Since Li is a complete space the sequence /„ must converge to another function in L 2 . This completes the proof. Appendix to Chapter 1 Certain theorems from real variables will be stated here for the reader’s convenience. The proofs can be found in Natanson p]. 16 Elements of Abstract Harmonic Analysis 1. Fatou's Lemma. If the sequence of measurable and non- negative functions /j, f 3 , • • - converges to the function F(z) almost everywhere on the set E, then j F(x) dx < supjjV»(*) dxj 2. Lebesgue's Theorem on Dominated Convergence. Let a sequence of measurable functions fi(z), /»(z), •••, converging almost everywhere to a function fix) , be defined on a set E If there exists a function H[x) summabte on E such that for all n and x \fn(x)\<H{x), then Iim j fn(x) dx ~ J f(x) dx 3. Fubini's Theorem. Let fix, y) be a summable function defined on the rectangle R{a < x < b, c < y < d) Then (1) the function /(x, y) considered as a function of y alone will be summable on £c, d] for almost all x G [a, 6], (2) if Q is the set of those x £ fa, 6] for which f(x, y) is summable on C c . dl, then the function g{x) = j J f(x, y) dy is summable on Q (3) the formula j Jf(x, y) dxdy = j dxj^ f(x, y ) dy is valid 1. Fourier Transform for Functions in L i 17 4. Tonelli-Hobson Theorem. If one of the iterated integrals J dxj f(x, y) dy, J dy Jf(x, y) dx converges absolutely, the double integral fff(x, V ) dx dy converges absolutely also, and all three have the same value. 4. Cauchy-Schwarz Inequality. If / (E L 2 and g 6 L 2 , then j I f(x)g(x) i <fej < | J | f(x) | 2 dx^J | g(x) | 2 dx . 5. Minkowski’s Inequality. If / € L p and g £ L p , then (/ \f{x) + g(x) \*dxy < I fix) | ”dxy + ( j | g{x) | p dx'j (p > 1). 6. Holder’s Inequality. If / £ L p and g £ L p where p and g are such that 1 + 1 = 1 V 9 and p> 1, then the product/g is summable and the inequality |/ l/(*)9(a ; ) I dx J < \ f{x) | p dxj (^J | gix) is valid. Exercises 1. Let {/„} be a sequence of functions in Lj(— 00 , 00 ) such that || /„ -/||,^Oasn-> °°. Then show that lim n ^. aj / n (a:) = f(x) uniformly on R. 18 Elements of Abstract Harmonic Analysis 2 Suppose/ € ii(— oo, «) Prove that lun J(x) = 0 »*±o> 3 Show that every point of continuity of / is a Lebsegue point of / 4 If/€ L p (— <» oo ) then prove that hm f {f(x + A) - f(x) \*dx = 0 A-0 •'--00 5 Let { /„) be a sequence in L„ such that lim,-„ {{/•—/ ll» =» 0 and such that/* » g Then show that / =» g a e 6 If / and g are continuous functions defined on R such that / ■= g a e then show that / = g 7 Let { U\ be a sequence in L } If hm,-*, ||/» — / IU * ® then prove for any g Li hm j f„{x)g(x) dx = f /( x)g{x) dx References 1 Zaanen Linear Analysis 2 Goldberg Fourier Transforms 3 Natansen Theory of Functions of a Real Variable CHAPTER 2 The Fourier Transform on the Real Line for Functions in L 2 In this chapter we will discuss the Fourier transform of functions in L 2 and also the inversion of such transforms. It will be shown that many of the results we had for Fourier transforms of functions in Li carry over to L 2 . Also the definition, several examples, and certain properties of Banach algebras are given. Fourier Transforms in t 2 In the last section we noted that / £ L 2 implied /„ £ L x n L 2 and that /„—>/£ L 2 and it is this limit function, f, that we shall take to be the Fourier transform of a function / £ L 2 . To paraphrase this we note that / has the property that lim ||/„- / 1| 2 = 0 n-*co or, one writes also f(x ) = l.i.m. n-*co Let us pause for a moment and see how this definition of f(x) matches up with our old one, denoted by Jo = /original if / £ Ll (1 Z> 2 . Suppose / £ Li n L 2 so that we are assured of the existence of /o, then fo(x) = J e izt f(t) dt = f e ix ‘ lim/„(f) dt ^ n-»co = lim J e ix, fn(t) dt r f(t)e' zl dt. 19 20 Elements of Abstract Harmonic Analysis by noting that f Z In and applying the Lebesgue dominated convergence theorem Thus /o = Iim /. Although, in general, if 9 =* 1 1 m g Mt this does not imply jr„(x) — * g(x) ae (i e pomtwise) we do have in this case (see Chapter 1, exercise 5) }•- i >t where / «= 1 j m as the connection between the two modes of computing the transform of a function in In n L, It will now be shown that a result we had for / Z I* f> In carries over exactly to the Case where f Z In Theorem (Parsecal). Let / £ Lj, then ll/ll, » V5 1|/ 1|. Proof Consider the sequence { /»( By definition of f we have hm | I/. - /Hi - 0 This implies! i™ II All. - 11/ II- « t This follows because just, as we have J| * J — 1 y ]J < ) * — V l * or absolute values we can show j ||/|) a — !! p |)» | < 1!/ — 9 1I» holds for the norm 2. Fourier Tronsform for Functions in Lj 21 lim|l/„l| 2 = li/lla (2) ft-* co and, since /„ £ Li n L 2 if / £ L 2 , we can apply the theorem of Chapter 1 (p. 13) to get II /-II*- VS II/- II* (3) Taking the limit in (3) as n — * =° and substituting (1) in the left-hand side and (2) in the right-hand side gives 11 / 11 ,-^ 11 / 11 *. We now turn to the problem of recovering the function from its transform or: Inversion in L 2 Theorem. Let/, g £ L 2 . Then ff(x)g(x) dx = jf(x)g(x)dx. Remark !./,{/ £ Lj, implies /, g £ L 2 . Remark 2. Each of the above integrals must exist by virtue of Remark 1 and Holder’s inequality. Proof. To avoid confusion we denote by /*(£) the function MO no, i*i<fc 0, I t I > k Mx) = j e ix ‘f k (t) dl Mx) = J dt. 22 Element* of Abstrocl Harmonic Analysis (Tilts tntko sett'e bocau«c/», ff» € L\ fl L, ) It non follows that //*(»)?«.(*) df Sinrt the wpiv^ioii oil the right com ergev abaolntcl} we can applj the 1 onellv-Hobson theorem and interchange the order of integration to get y*/»(x)?«(x) dr ~ ffk(t) rt(fg.{x)t“'di or ffkix)g.{x) tlx » 1) <tt (4) Wt nni-t now note two farts and the revolt will be proved Fird it iv setn that hm (| v. - jr ||, « 0 =* Urn |l (/. - g ||, - 0 beeaus*. II if* ~ t» lit - I! ?• - 9 1! from I\r^\alv theorem Second if a sequence of function? {A. I coimrgev to h m the norm thl> imphe. Inn j k y (T)k(z) sir ” j h{x)k(x) dr for all k(x) m L» (<ee Chapter l ext rose 7) With the«e two remit > m rouid ht u> take the limit as » -* *e of each side of (41 lww ** & 2. Fourier Transform for Functions in I 2 23 or ) dt = Jfk(t)g(t) dt. It now remains only to let k become infinite. Before proceeding to our next result which is the inversion formula it is advisable to note that / will mean transform f first and then take its complex conjugate. Theorem. Let / £ L 2 and let g = /, then / = (2t)~ 1 p. In words this says: if you start with the conjugate of a function’s transform and you then transform it and conjugate it, dividing by 2rr yields the original function. Proof. If we can show ll/-^lk = o, 2ir then we will be done. Omitting arguments, we consider We now observe that / fg = Jfg = /// = l[/[li = ||/ II2- 24 Elements of Abstract Harmonic Analysis Now, since II 5 115 - 2ir||g||5 - 2r||/||i ■> 4ir> ||/||1, 11/ - ^ III - 11/115 - ll/tli - 11/115 + ~ 11/115 - o QED As a corollary we can now state the following Invertlon Formula. If f £ L*, then m = 1 1 m — * r dx 2tr Proof Let f € Li and let / — g Then ,m £? 8nd } -i s More precisely /(t) “lira ~ f e ,t( )(x) dx Taking the conjugate of both sides, f(t) = 1 i m — f e"**'/(x) dx n-d> 2ir We can now state the following result Theorem (Plancherel). Let / € Then there exists a func- tion, / € Li, such that /(*) = hm f e ul f(t) dt and f(0 =1 lm ~ j (T^'jix) dx, also ll/H* = (2 t)'« H/ll, 2. Fourier Transform for Functions in 1 2 25 As a consequence of this inversion formula and Plancherel’s theorem we can now assert that every function in L 2 can be viewed as the transform of another function L 2 . We note that if two functions have the same transform they must be equal almost everywhere by the inversion formula and will be considered as actually the same function. We can summarize this by saying that the mapping of L 2 into L 2 by the Fourier transform is both 1-1 and .onto. Normed and Banach Algebras Definition. A set X is called a normed algebra over C, where C is the field of complex numbers, if (1) X is a normed linear space over C, (2) X is a ring with respect to two internal operations, the additive operation being the vector addition in (1), (3) k is a member of C and x and y are vectors in X, then k{xy) = ( kx)y = x{ky). Thus we have established a link between the external and internal multiplications. (4) The norm must have the property that, with respect to internal or ring multiplication, II x v II < IMI-IMI. x >v € x. If, in addition, A is a Banach or B-space ( complete normed linear space) then X is called a Banach algebra. To clarify the concept of a Banach algebra several ex- amples will be considered. In most cases the examples will clearly be linear spaces and all we will do is to define a multi- plication and a norm that satisfies property (4). One final word of warning about (4) is in order though: the product intended on the left-hand side of the inequality is meant to be the ring product or product with respect to internal multiplication. 26 Efemenfi of Abstract Harmonic Analysis Example 1. LetAf be a Banach ■space and consider L(X, X), the class of all bounded linear transformations on X (A linear transformation A on a normed linear space is said to be bounded if there exists a positive real number, K, such that f J A (x) (| < K JI z f| for all x m X ) We define multiplication of bounded linear transformations in the natural way (AB(x) = A (B(x) ) ) and take as a norm on L(X X), II A || - sup II Example 2 Let X Cfa 6}, complex-valued continuous functions on fa, 6] W e define multiplication in the point wise fashion and take as the norm of / € X the maximum value that the absolute value of / attains on the closed interval or |!/|| = max \/(x) I «uu Example 3 Let TT be the set of all absolutely convergent trigonometric «enes _ c.e tn 1 and take as the norm of any *(t) m It 11 *(0 II “ 11 It c.e"' H = £ I c. 1, with multiplication taken as the Cauchy product Example 4 Let A consist of all functions analytic in the open unit disk in the complex plane and continuous in the closed unit disk We «haU take as the norm of a function, /, in A ll/ll = max|/(z) 1 = max |/(z) |, i«i<i i *i-i and multiply such functions pointwise 27 2. Fourier Transform for Functions in L 2 Example 5. Let the space P n+1 denote the space of all polynomials of degree less than or equal to n. We must first define multiplication of two such polynomials in a way that is closed. To this end we take h(t)g(t) = t=o where n n C* = E a M, Hi) = E a / y > and g(t ) = }+l=~k j = 0 1=0 Next, we define || aft || = E"=° I “i I- Example 6. Suppose Lj is the space considered in Chapter 1 with multiplication defined as fg = / *g- We showed in Chapter 1 that Example 7. Suppose G = jci, o- 2 , m„} is any finite group. Consider the class of all complex-valued functions on G ; i.e. all / such that /: G —> C. This class is denoted by Li(G) and is called the group algebra of G We will denote multiplication in Li(G) by * and define the product of two functions/, g in Li((7) as follows: = E /Ws (°v)- ffioj—ffk Now, since we can write <r; = o- t cr ; - _1 , we have (f * g) (<r*) = E"-i/(‘ r * ff r 1 M<r/). (A careful inspection of this process reveals that it is, formally, similar to the usual multiplication of polynomials.) As a norm on Li(G) we shall take 11/ 1! = E !/(-••) I i=I 28 Elements of Abstract Harmonic Analysis and we shall actually demonstrate here that this norm satisfies (4) Consider II/-SII - El </•«)(«> I “ Si I2/(<r*«rr I )ff(<r J ) [ t i ^ H Si I * i * ]£ I ?(«■/) I 2 l/(wr*) I -ll/ll II All Example 8 Let £ denote the set of all integers, and take Li(Z) to be all complex valued functions, f, on Z such that fi W»> \ < w Let us define multiplication of two functions / and g from Li(Z) / * g as (/•!!)<») - E /(» - and take the norm of / to be ll/ll- Ei/wi Without verifying the claims we assert that this too forms a Banach algebra Examples 6 7, and 8 have many features in common each had a multiplication that resembled (or actually was as in the case of Example 6) convolution Actually through the use of an abstract integral we could have defined multi- plication in Examples 7 and 8 as convolution also by ap- propriately assigning a measure to the set of all subsets of the space The similarity extends beyond this, however 2. Fourier Transform for Functions in I 2 29 The space underlying the function space was in each case a group but more than that each space was actually a topo- logical space, the discrete topology being assigned in the case of Examples 7 and 8. But there is still more that each had in common: Each is a locally compact topological group. We leave these observations now to pursue something else that is more immediate but will return to them again, later. Analytic Properties of Functions from C into Banach Algebras We now turn our attention toward mappings from the complex numbers, denoted by C, into Banach spaces, X, over C. Symbolically, if D is a region in C, then let x: D->X X — *• x(X). It now makes sense to make the following: Definition. The function, x(X), is said to be analytic in D if x'(Xo) = lini X-*Xo s(X) — x(Xp) X - X 0 exists for all X 0 in D. Note here that the limit is taken in the sense of the norm on X. Suppose we consider the mapping x f CD D-^X-rC where /is a linear functional on X has the property that there exists a k > 0 such that |/(x) | < fc jjxjj forallx€X; / is called a bounded linear functional on X. Here / is a complex-valued function of z £ A" and }x is just a function of the complex variable, X. Denoting the class of bounded linear functionals on X by X we state, 30 Elements of Abstract Harmon c Analys without proof that sup 11*11*0 l/M I II *11 is actually a norm on X and we write l/M I 11/ II - sup lull We also state without proof the following Theorem A linear functional is bounded if and only if it is continuous In view of these facts suppose / X — C and / € \ If x(X) is analytic in D then /(*(X)) is analytic in D To prove this consider 1 m /l<W) -/(*!>.)) _ A »(>) - IQ..A X — Xo X — Xo / -f(x (Xo)) by using linearity and the fact that boundedness is equivalent to continuity which allows us to interchange the order of computing the functional value and taking the limit M e will now prove Liouville s theorem for vector valued func tions of a complex variable Theorem Let x C — * X where X is a Banach space If i(X) is analytic m the entire plane and bounded (le II *(X) H < M where M > 0 for all X € C) then x(X) must be constant Proof As noted above /(x(X)) is analytic when / £ ^ Since/ € ^ we can say there exists a l > 0 such that 1/(*(X))| < Ml *(X) || <kv 2. Fourier Transform for Functions in L 2 31 But since fx is just a complex-valued function of a complex variable that is also entire and bounded we can apply the ordinary Liouville theorem to it and assert that / must be constant. Let a and /3 be any two complex numbers. Then /(*(«)) =/(*(«) or, by linearity, f(x(a) - x(fi)) = 0. But / is any bounded linear functional on X. Therefore, since every bounded linear functional vanishes on the vector x(a) — x(/3), it follows from a consequence of the Hahn- Banach theorem, that the vector must be zero or x(a ) = x(P) for any a, p which is equivalent to saying x(\) — a constant for all X. We can now extend another familiar result. X Theorem. Let X be a Banach space and let C 3 D > X. If x(\) is analytic in a region bounded by a rectifiable Jordan arc F and continuous on T, then f x(X) d\ = 0. •'r Before proceeding with the proof we make the following two observations: 1. The integral here is defined as a limit in the norm. 2. The integral must exist by virtue of completeness of space and the continuity of x(X) on the contour itself. Proof. Let / € X and let y = Jrz(X) dX. Reasoning as in the previous theorem the following manip- ulation is justified if / € X : f(y ) = J /(z(>0) dx. 32 Elements of Abstract Harmonic Analysis But /(*(A)) is a complex-valued analytic function in the region and continuous on r Therefore by the Cauchy integral theorem /(«) - 0 As before since every bounded linear functional vanishes on y y must be zero or y - j x( A) dX = 0 The following result will not be proven but follows simply from the above as m the classical case Theorem Under the same hypothesis as the preceding theorem *(& 1 f *(A) , We could also show using this that x'(*e) existing im plies x (X 0 ) exists and so on and also generate a power series expansion about X 0 for x(X) with radius of convergence the power series being x(X) = fX — Xo) n , where We now focus attention on some of the internal properties of Banach algebras Theorem Suppose X is a normed algebra then nng multi- plication is a continuous operation 2, Fourier Transform for Functions in Lj 33 Proof. Let x , j/, x q , j/ 0 6 X. Then II *2/ - x 0 y 0 || = II (* - xo) ( y - yo ) + x 0 (y - yo) + {x - x 0 )y 0 || ^ II x ~ xo || || y - y 0 1| + || x 0 || || y - yo || + II 2/o II II* - 2o||. Taking the limit as x, y — » x 0 , yo yields the desired result. Theorem. Let X ^ |0J be a Banach algebra with identity, e ; i.e., ex = xe = x for all x £ X. If x £ X and || e — x || < 1, then: 1. x is a unit of X (i.e., x has an inverse) and 2. 2 T 1 = e + Z"( e - *)”■ Remark. The summation Z”( e — x)” exists (in the norm) for II (e - x) n || < || e - x ||" < 1. Proof of Theorem. Write x — e — (e — x) and take (e — (e — x))(e + Z( e — x)") = e -f (e - x) 1 + (e — x) 2 + ••• — (e — x) — (e — x) 2 — ••• = e. QED Exercise Let X be a Banach space. Suppose y,"_iXn converges abso- lutely, i.e., yi”_, || x„ || converges. Then show that converges, and || || < Z II *» ||. References 1. Naimark, N armed Rings. 2. Gel’fand, Raikov, and Silov, Commutative Normed Rings. Amer. Math. Soc. Transl., No. 5. 3. Taylor, A. E., An Introduction to Functional Analysis. 4. Kolmogorov and Fomin, Functional Analysis, Vol. 1. CHAPTER 3 Regular Points and Spectrum In this chapter we continue with our study of Banach algebras and briefly introduce the Gel’fand theory of com- mutative Banach algebras We state and prove the following Theorem 1 Let X be a Banach algebra with identity Let X be a complex number such that |X[ > ||*||, then x — Xe where e is the identity of X, is a unit Proof First we note that if x — Xe is a unit, then Xe — * is also Since x — Xe = X(X 'x — e), then if we can show (X -, x — e) is a unit we will be done To this end consider the quantity (e — X -1 x) Taking the norm of e — (e — X l x) gives which is less than one by hypothesis Using the theorem on p 33 then e — X ‘x is a unit which implies (x — Xe) is a unit We now note that Xe — x >H) 34 3. Regular Points and Spectrum 35 = 5 >-». X "- 1 7J — 1 or (Xe — x)- 1 = '%2\-’ , x n - 1 . n= I Example 1. An immediate application of this theorem is available to those familiar with functional analysis: Let X be a complex Banach space and let L(X, X) denote the class of all bounded linear transformations mapping X into X. The set L(X, X) is a Banach algebra with identity, f where we take the norm of A € L(X, X) to be || A !! = sup * 5*0 jUM II 11*11 ' By the theorem, if X is a complex number such that 1*1 > IIAII, (X — A) -1 exists and (X — A)~ l = y^,”-.X~’ 1 A"" 1 . Let X be a Banach algebra with identity, e, and call the set of all units of X, U. Symbolically U = (a: 6 X | x exists). f See Taylor [1] for a discussion of this. 36 Elements of Abstract Harmonic Analysis It is noted that certainly e 6 U Consider a one-neighbor- hood of e, Si{e), le , &(e) - {*€*(||e-*||<lj By the theorem on p 33, St(e) C V Now, if i ^ 17, then xx~ l = e Since ring multiplication is continuous (see p 32) there must exist a neighborhood of x, N{ x) (in the norm) Buch that the set N(x)x ~ > C S t (a) where N(x)ir' = (l»-‘ I y ( N(x) | Let y € N(x) Then V*r' e Stic) C V which implies yi -1 is a unit also This means that there exists an element z € X such that (yx~ l )z = y{x~'z) =» «, and zyx~‘ = eor (ar*z)y = e This nnplies that y € V But y was any pomt in N(x) and x was any element of V so that every pomt of U is contained in a neighborhood that lies wholly within U, therefore U is an open set Thus the set of units, U, is an open subset of the Banach algebra X Theorem. The mapping / U —* V given by f{x) — x~ l is continuous Proof All we need show to prove this is that if the sequence i n — * x, then I* -1 — ► i -1 Let 1 1 «) be a sequence from U such that — » x (in the norm) This implies x~ x x« —* e or for any < > 0 there exists an X(e) such that ii *-■*. - « ii < « for n> N 3. Regular Points and Spectrum 37 Choose Ni such that || x 1 x n c II ^ 1 for tt Ai and consider the series e + Y( e - a r l x n ) k for n > Ni. 1 By the theorem on p. 33 and the fact that || x l x n — e || < 1 we assert that the series converges absolutely to ( x~ x x „ )-’ = x -ix. From the absolute convergence, then, || e - x^x || < Y || (e - at-'ain) 1 ' || = Y !l x ~ k ( x ~ x ^ k II *=1 < £ II s" 1 11*11 (*-*»)ll‘- (,) Jt=l Since x -> x n or || x - x„ || -» 0 we can choose N 2 such that n > N 2 will make (*) as small as desired or || e - x~'x || — > 0 which means x~ l x — > e or a:” 1 — > ar 1 . QED Before proceeding further two definitions will be stated. Definition 1. Let X (E C. If x - Xe is a unit, then X is called a regular point of x. Definition 2. The set of nonregular points of z is called the spectrum of x and will be denoted by <r(x) . Thus X x , then (a; — Xe) $ U. Example 2. To those familiar with functional ana ysis following fact is apparent: Let L(X, X) be t e aMC " algebra as in Example 1, and let A 6 L(X, A). , is « regnkr point of X. then (X - A)- € h(X, X) wh.oh implies X € p{A) where p(A) is the resolvent se o 38 Elements of Abstract Harmon C Analysis Suppose conversely that X 6 p(A) Then (X — .4) 1 must be bounded and the range £(X — -4) must be dense in X Tie would like to show now that X must be a regular point of x and for this we ask the reader to recall the following two theorems from functional analysis f (1) If a linear transformation is bounded on all of the Banach space Y then it is a closed linear transformation a (2) Suppose XD D-t K If X is a Banach space and B a closed linear transformation such that is bounded on the range B(D) then B(D) is closed In view o f these two theorems we can say that R (X — A ) = R(h — A) ( the closure of R(X — A)) But ft(X — A) was dense in X or R(\ — A) - X Therefore R(\ - A) * X This says that (X — A) is bounded and defined on all of X hence (X - A) 1 € L(X X) which implies X is a regular point Thus we have p(A) - {set of regular points of A} Taking complements we obtain C(p(A)) — <r(A) {usual definition of o-(A)} = tr(A) {as given m Definition 2} Hence our old notion of spectrum (le the functional analysis notion) agrees with that given m Definition 2 Compactness of the Spectrum Let X® be a regular point of x This means that xx, = (x - \fi) € V t See Taylor [2) 3. Regulor Points and Spectrum 39 Since U is an open set, there exists a neighborhood of x Xo , iV(x x „), such that iV(x Xt ) C u. Consider the function y\ = x — Xe; i.e. 2/x: C->X X — > x — Xe. This function is continuous for, certainly, X — ■» X 0 implies (x — Xe) — > (x — X 0 e) . Since it is continuous, given a neighborhood of x x „, iV(x Xo ) C U, there must exist a neighborhood of X 0 , A 7 (Ac>) C C, such that yx e N(x\ 0 ) c u for X € iV(Xo). Thus, y\ € U. In view of this, all the points in iV(X 0 ) are regular points; hence the set of regular points is an open set because every point has a neighborhood lying wholly within the set of regular points. This implies that <r(x) is a closed set in C. We already know by Theorem 1, that if | X | > || x ||, then (x — Xe) _I exists which means that c(x) must be con- tained in the closed disk of radius || x ||, or X 6 <t(x) => | X | < || x ||. Therefore o-(x) is closed and bounded. By the Heine-Borel theorem for the plane, then cr(x) is a compact set. Using the definition of analyticity given in Chapter 2 we would now like to show that the function x(X) = (x — Xe) -1 . 40 Elements of Abstract Harmome Analysis where X is regular is an analytic function of X over the set of regular points To do this we will first obtain a certain equality Consider the function i C*->X X -» (x - Xe) 1 where C x C C is the set of regular points and x is some element of X (Note we are identifying the element x with the function x ) Let Xi and Xj be regular points and take ®(Xi) *x(X*) ~ (x ~ Xje)x(Xj) «=((* — Xj«) + (X* - Xj)e)x(Xj) - (x - + (M - Xi)x(X,) = e + (Xj — Xi)x(Xj) Thus z(\ t ) - x(X t ) + (A* - or x(X 3 ) - x(Xj) = (Xi - XOxfXOxfX,) Theorem x(X) = (x — Xe) ‘ is analytic in the (open) set of all regular points Proof LetX X 0 be regular points By the preceding estimate Urn =, h m x(Xo)x(X) (1) x-x» X — Xo x-x 0 Now we note that the function x — Xe is a continuous func tion of X Second it is recalled that in a Banach algebra x„ — * x => x„ 1 — * x~ i 3. Regular Points and Spectrum 41 and third since, by continuity, lim x — Xe = x — Xoe, x-x 0 then lim ( x — Xe) -1 = (x — X 0 e)~K X-Xo Hence the limit, (1), is just ( x — Xoe) 2 . Example 3. With L(X, X ) as in Examples 1 and 2, let A £ L(X, X), X € p{A), and let R\ denote the resolvent operator, i.e., R\ = (X — A)~ r . Now |Ex = |(X-^)- 1 =-|M-X)-> = -E 2 Theorem. Let Xbea Banach algebra with identity let x £ X. Then o(x) <f>. Proof. Suppose Then cr(x) = 0. x(X) = (x — Xe) -1 is analytic throughout C. But e — x X 11 II — > e ii ii as | X | — » oo which implies (e — (x/X)) _1 — » e -1 = e as | X | — x «> . Now |i (at - Xe)" 1 1| = | X- 1 1 || (e - X-x)- || -» 0 as | X | -^ co (2). Therefore (x — Xe) -1 is a bounded, entire function. By the form of Liouville’s theorem proved in Chapter 2, then (x — Xe) -1 = a constant 42 Element* of Abstract Harmonic Analysis and by (2) above, the constant must be zero But this is nonsense, for if (j — Xe)" 1 = 0 how can (x — Xe)(x — Xe)" 1 — e ? Thus the assumption, o{x) — 0, has led to a contradiction and therefore »(x) ^ 0 QED We will now state and prove an important theorem about the structure of certain types of Banach algebras Theorem. If X is a complex Banach algebra with identity and, in addition, if X is also a division algebra, then X is isomorphic to C Proof Let x 6 X Knowing that &(x) ^ 0, let X 6 <r(x) Consider {x - Xe) Since X £ <r(x), (x - Xe)- 1 does not exist But we are in a division algebra and all elements have inverses except zero Hence (x — Xe) = 0 or x — Xe Now we have it, given any x f X, it is equal to 6ome scalar multiple of identity Thus we have the rather natural mapping from X — > C where the image of any x ^ Xe is taken to be X or X —> C where Xe — » X It is exceedingly simple to verify that Xie X*e — * X% Xi XieX 2 e — > XjXj and if ct £ C that aXe — » aX Further, the mapping is clearly 1-1 and onto which establishes the isomorphism We note that the mapping is an isometry if ||el| = 1 Some new definitions are now necessary Definition. The real number r r (x) = sup [ X | M*) is sniff tn hp flip x-nertrnl rndiun of x 3. Regular Points and Spectrum 43 Remark. Since | X | > || x || implies X $ <r(x), then certainly r,(x) < || a: ||, i.e. <j(x) C M where M represents the closed circle shown below. Before proceeding to our next result we will need the following: Lemma ( Special case of the spectral mapping theorem ). Let X be a commutative Banach algebra with identity, then <r(x n ) = o-(x)", where x 6 X. Proof. Suppose first that X 9^ 0 and X € <r(x n ) and let cui, a 2 , — , u>„ be the nth roots of X. Now x" — Xe = (x — cure) • • • (x — oj„e) . but at least one of the factors on the right must be non- invertible. Suppose it is (x — u>,-e) . This means that w, 6 <r(x) which implies cu” € a(x) n or X € c(x) n . Therefore o-(x") C <r(x) n . (3) u Elements of Abstract Harmonic Analysis On the other hand, suppose a (E <r(x) B and let ft, ft, • ■ ft be the nth roots of a, so that ft £ a(x) for some i Now consider (x — fte) ( x — fte) •••(« — fte) = x B - ae If x “ — ae was invertible, then we could multiply by (x B — ae) -1 to get (x„ - ae) _1 C(* - fte) • • • (x - fte) 3 *= e Since X is a commutative algebra we can write (x» - ae) -1 £(x - fte)--*(x - fte)3(i - fte) = e which says x — fte has an inverse Therefore x“ — ae is not invertible and a € <r(x") or o-(x)" C <r(x") (4) Finally if 0 6 a (x n ) , then x" is not invertible and therefore x is not invertible, so 0 6 v(x), whence 0 € a(x)* Con- versely if 0 £ <t(x) b , then 0 £ <r(x) so x is not invertible, consequently x B cannot be invertible so 0 £ <r(x B ) Combining this with (3) and (4) we have <r(x“) = <r(x) B QED Theorem. Let x £ X (usual assumptions about X) Then r,(x) < hm || x B |j 1,B 3. Regular Points and Spectrum 45 Proof. We first note that since <r(x n ) = <r(x)”, then T,{x n ) = sup I X I = sup I X I = sup I n I" = / sup I ft I Y* Xecr(x n ) Xe<r(x) n peff(x) l pftr(x) } ■ Thus r <r (x n ) = r,{x) n . \ Since r„(x n ) < 1| x n || we also have (r c (x)) n < 1| re" |l or r,(x) < || x n || 1/n for any n. Therefore r„(x) < lim || x " |l I/n . QED n Since lim || x " | | I/n < lim || x n || I/n n n if we can show t c {x) = lim || x" H l/ ", n we will have r,(x) = lim || x n [| 1,n . n-^co To this end consider the following: Theorem. Under the same assumptions about X, let x £ X. Then: (1) If | X | > r„{x), then (Xe — x)~ l = EX-"*- 1 . (“) 1 (2) If the series, (a), converges for | X | = t c (x), then it represents (Xe — x) -1 . (3) The series (a) diverges for j X | < r„(x). 46 Elements of Abstract Harmon c Analys : Proof LetX be a regular point and let ( X | > r»(x) (See Fig 1) 44 (1) By the definition of r,(x) (Xe - x) 1 must exist for | X | > r,(x ) By the theorem on p 40 (Xe — x) 1 is an analytic function throughout the set of regular points and therefore (Xe — x) * is analytic for | X | > r„(x) Analyticity however implies the function possesses a untque Laurent expansion Thus the expression ~ H x* 1 for (Ac — x) -1 for ] X | > || x || must agree with the expres- sion for the inverse for | X | > r.{x) or (see p 35) (Xe - x) 1 - £x "i" 1 i X | > r,(x) (2) Suppose (a) converges Then we contend that it represents (Xe — x) -1 for <>e - x) 1 - -+‘I- 1 - £X-*Z* 3. Regular Points and Spectrum 47 Changing the index of summation in the first term yields (Xe — x ) = e -J- ^\~ n x n — ^X~ n x n • i i i = e which establishes (2), and also (3), namely: (3) If X 6 <r(x), then (a) must diverge. Suppose | X 0 1 < r„(x) and suppose (a) converges at X = X 0 . If (a) converges at all it must, by the preceding part, converge to (Xe — x) -1 . Since (a) is a series in 1/X, then if it converges for Xd it must also converge for j X | > j X 0 1. But there must exist Xx £ <r(x) such that | Xi | > | X 0 | and (a) must converge there. Thus the assumption of convergence at a X 0 where | X 0 | < r„(x) has led to contradictory results. Therefore («) diverges for | X | < r„{x) or 7v(x) is the radius of convergence for (a). QED We note that the power series in 1/X must have lim |J x” || 1/n n as radius of convergence bj r the Cauchy-Hadamard formula. Therefore r„(x) = lim |! x" || I/ ". 71 Combining this with the theorem immediately preceding the above theorem we have r„(x) = lim || x” || 1/n n-*co 48 Element* of Abjtract Harmon c Analysis Introduction to the Gel’fand Theory of Commutative Banach Algebras At this point we will introduce the Gelfand theory of commutative Banach algebras In the following discussion X will be assumed to be a commutative Banach algebra with identity e and further it will be supposed that the norm of e is one Definition A subset / of X is said to be an ideal if (1) J is a subspace of X and (2) x € X y £ / implies ry £ I Remark It is easily verified that the above conditions are equivalent to the following two conditions (1) x y £ 1 implies (x + y) £ / and (2) i € / a € X implies zx £ I Definition An ideal is said to be proper if there exists at least one element in X that is not in the ideal Theorem If / is a proper ideal in X then I the closure of / is a proper ideal in X Proof First it will be shown that I is an ideal in X and then it will be show n that if / is proper then / is also (a) I is an ideal Let x y £ / There must exist sequences |z,} and |y*} such that Zn — * x and y»—*y where x. € f for all n Since / is an ideal (z„ + y n ) £ I for all n Since this is true for all n then the limit of the sequence (x n + y»! is in I, or (* + y) € / f»ow let z £ X and let x„ — * x where x £ / Since zx* £ I for all n it follows that zx £ I Thus I is an ideal (b) / is proper Let z € / If x~ l exists then x~ l x £ / or e £ / Since multiplication by any element is allowable this would say that I = X Therefore if x £ / then ar 1 cannot exist In 3. Regular Paints and Spectrum 49 other words proper ideals must be made up solely of singular elements (i.e. nonunits). Letting U denote the set of units of X as before then we now have, denoting the complement of U by CU, This implies I C CU. ZC CU. Sinc e U is an open set, CU is closed which means CU = CU or ZC CU. Now, since e | CU, e $ I. Thus Z is a proper ideal which is the desired result. Before proceeding another definition is necessary. Definition. An ideal, M, is called maximal if the only ideal that contains M properly is all of X, or i f J D M properly where J is an ideal in X, then J must equal X. Theorem. (1) If Z is a proper ideal, then there exists a maximal ideal M such that M C Z. (2) If x is a singular element, then there exists a maximal ideal that contains x. Proof. This result is a standard result from modern algebra and in it we will not have to make use of the fact that X is a Banach algebra, the result being true for any algebra with an identity. (1) Let S denote the set of all proper ideals which contain Z. It is easily verified that set inclusion induces a partial ordering on .S’. Let T = {Z„J be any totally ordered subset of S. If we can show now that T has an upper bound that is also a member of S, we will have shown that S is induc- tively ordered and can apply Zorn’s lemmaf to assert the existence of maximal element for S. Clearly an upper bound for T is just U a I a . We must now show that U a Z a is also a t Zom’s lemma: Every inductively ordered set has a maximal element. 50 Elements of Abstract Harmon c Analys proper ideal In general the union of ideals is not an ideal but since T was totally ordered it follows that U „I a is also an ideal and it is proper since e $ U«I B Lastly since every I a contains 1 it follows that U a / a is a member of S Therefore S is inductively ordered with respect to set inclusion and by Zorns lemma must have a maximal element which is clearly a maximal ideal Thus there is at least one maximal ideal that contains I (2) Let x be a singular element and consider the principal ideal (x) - {xy | y € X} Pt is simple to verify that (x) is actually an ideal 3 We note that e cannot belong to (x) for if it did there would have to exist a y € X such xy •= e which would say that x is a unit contrary to our assumption Further we see that ex = x € (x) so that (x) is nonempty Thus (x) is a proper ideal in A and we can apply part (1) of the theorem to assert that there exists some maxima] idea] containing (x) and hence also containing the element x QED We now note that any maximal ideal must be closed for as we noted earlier M being a proper ideal implies M is a proper ideal too In any case M Q M and if the inclusion were proper this would contradict the maximahty of M Therefore if M is a maximal ideal M = M We will now show that given any maximal ideal m X there is another Banach algebra immediately available The Quotient Algebra With the same assumptions about X let M be a maximal ideal in X It immediately follows that M is a closed sub space in X We now claim without proof that the quotient algebra with respect to M X/AI is also a Banach algebra commutative with identity A typical element of X/AI is the eoset x + M u here x is an element of X Addition of cosets is defined as (x + M) + (y + M) - (x + y) + M 3. Regular Points and Spectrum 51 multiplication as (x + M) (y + M) = xy + M, and scalar multiplication as, if a is a scalar, a(x -f- M) — ax + M. As a norm on X/M we take \\(x + M) |! = inf || y ||. X/ex+AT It is to be noted that the faet that M is closed is essential here, if what we have called a norm is to be truly a norm for we desire the property that || a: + M || > 0 and equals 0 if and only if x is such that x + M is the zero element of X/M. It is here that M being closed is essential. For our next result we will need the following theorem from algebra. Theorem. If R is a commutative ring with identity and M a maximal ideal in R, then R/M is a field. (The reader is referred to any good book on modern algebra for a proof of this; e.g., Van der Waerden [230 By this theorem then, using the fact that X is a commuta- tive ring with identity, we can say that X/M, in addition to being a Banach algebra, is also a field. By the theorem of p. 42 it now follows that X/M is isometrically isomorphic to C, the complex numbers. Since, by the same theorem, it follows that every element of X/M is some scalar multiple of identity; i.e. given x and M there exists a scalar x{M) such that x + M = x(M)(e + ill), the isomorphism can actually be exhibited by the following mapping iso X » X/M » c (a) x — * x + M = x(M) ( e -j- M) — *■ x(M). 52 Elements of Abstract Harmon c Analys : But there is something else going on here as well If we choose some particular element X we can map the set of all maximal ideals M the set of all maximal ideals m X into the complex numbers via x M C M -* x(M) where we have identified the element i f X with the func tion x M —» C We now claim the following assertions are true for mappings of this type Theorem (1) If x — y + z then x(M) - y{M) + z(Af) (2) If x - ay where a is a scalar then x(M) - ay{M ) (3) If x - yz then x(M) = y(M)z{M) (4) e{M) - 1 (5) x(M) — 0 if and only if x belongs to M (6) If M N are distinct maximal ideals then there exists an x € X such that x (M) ^ x(N ) (7) | z(M) | <||*H Proof Statements (1) (4) all follow directly from the fact that the mapping (a) above is a homomorphism from X into C (5) x(ilf) + AT = 0 + <=>*€** (6) If M y* N then there exists an element x that belongs to M but not to N Therefore x(Af) = 0 but x(N) 9* 0 (7) Write *-f &f- z(Sf)(e + Jff) By the definition of || x + M ||, 11*11 > II* + M|| * \\x(M)(e + M) || = 1 x(M) | || e + M ||. (b) But || e + M || < || e || = 1. Suppose the strict inequality prevailed; i.e. suppose || e + M || < 1. This implies that there must be a t/ 6 e + M such that II y II < i. If y £ e 4- M, however, it can be written y — e + x for some x £ M. But y=e+x=e— ( — x). Now since 1 > II 2/11 = lie - (-*) ||, this implies (— x) is a unit by the theorem on p. 33. This is impossible though because M is proper. Therefore II e + M || = 1.. Substituting this in (b) gives |l * || > I x(M) I || e + M || = | x(M) | which proves (7). Exercises 1. Let X be a complex normed algebra which is also a division algebra. Prove that X is isomorphic to C. 2. Let X be a Banach algebra and let x 6 X. x is called a topological divisor of zero if there exists a sequence [x n \ in X such that || x„ || = 1 and either xx n — » 0 or x n x — > 0. If i is a topological divisor of zero, show that x is not a unit 3 Let X be a Banach algebra Show that if 0 is the only topological divisor of zero in X, then X is isomorphic to C 4 Let X be a Banach algebra If there exists a constant m > 0 such that )) xy |) > m JJ * || )| y )| for all x, y 6 X, then show that X is isomorphic to C 5 Let Y be a closed subalgebra of the Banach algebra X Let x € Y and denote by ax{x) and ay(x) , the spectrum of x in X and V, respectively Show that (a) <rx(x) C *r(x) (b) each boundary point of trr(i) is also a boundary point of ax{x) 6 If X is a commutative Banach algebra and if x, y € X, show that r,(xy) < r,(x)r,(y) References 1 Toyicv A i. An Jnlraduelttm to Fvnthevusl Analyst* 2 Ian der Waerden Modem Algebra CHAPTER 4 More on the Gel'fand Theory and an Introduction to Point Set Topology In this chapter we will continue our discussion of the Gel’fand theory of commutative Banach algebras and will then briefly review some aspects of point set topology that will be necessary in the later development. Throughout the following discussion X will denote a com- mutative Banach algebra with identity, e, where the norm of e is equal to one. Theorem 1. Let x £ X and let M € M, where M is the set of all maximal ideals in X. Then a; is a unit if and only if x{M) 0 for any M € M. Proof. It is first noted that a; is a unit if and only if x is not a member of any maximal ideal. This is justified by noting: (1) if a unit belongs to an ideal, the ideal cannot be proper, and (2) any singular element is contained in some maximal ideal (see p. 49) . Further, x{M) = 0 if and only if x £ M . Combining these results with part (5) of the theorem on p. 52 implies x is a unit if and only if x(M) 9^ 0 for any M 6 M. QED. Our next result yields a new way to characterize the spec- trum of an element. Theorem 2. Denote the spectrum of x by <r(x). Then <r(a 0 = [x(M) I M € M}. Proof. Let x(M) = X where M G M and consider (x - Ae) (M ) : (x - Xe) (M) = x(M) - \e(M) 55 56 Elements of Abstract Harmonic Analysis but, since e(Af) =■ l, (x - Xe) (HI) = X - X « 0 <=> (x — Xe) £ ill <=> (x — Xe) is not a unit Therefore X € *(x) or {x(ill) J M e AT} C <r(x) Conversely, suppose X € «r(x) This implies x — Xe is not a unit By Theorem 1, then, there must exist an M € M such that (x - Xe)(HI) = 0 or x{M) - X = 0 or x{M) = X Theorem 3. Let x £ X Then hm I) x“ || ,,B ■ sup | x(AT) | » AUft Proof. As proved in Chapter 3 r,(x) — Inn |f xr ||"" = sup | X | = sup | x(M) | by Theorem 2 Mtfi Definition. The set is called the radical of X Remark. Since 0 £ M for every M £ HI, then ccrt&wly o e n jr«ftAf 4. Gel'fcmd Theory and Point Set Topology 57 Definition. If (1 mXiM = {0 j , then X is called semisimple. We 'now observe that x £ f) M <=> x(M) = 0 for all M £ M MeM <=> sup | x(M) | = 0 Mail <=*■ lirn || x ” || 1M = 0. n-+ca Elements, x, with the property that there exists an n such that x n = 0 are called nilpotent elements, while if lirn || x n || ,/n = 0, n-» cd then x is called a generalized nilpotent element. Having defined the set M we can now speak of mapping it into other sets. In particular we would like to consider mappings of M into C and we will denote the class of all functions that map M into C by F{M) ; i.e., F(M) : all complex valued functions on M. Certainly, functions of the type mentioned at the end of Chapter 3 are in this category where we identify the elements of X with the function x: M->C M-^-x(M). Identifying the elements of X with these functions defined over M can be viewed as the mapping (where, in this case, we denote the function x by its functional value x(M)) X -> F(M ) x-*x{M). (*) It is clear that F(M) is actually an algebra with respect to the usual operations of addition and multiplication of functions and scalar multiplication of a function by a com- plex number. With this in mind it is now claimed that the 58 Elements of Abstract Harmonic Analysis last mapping, (•), is actually a homomorphism from X mto F(Af) This follows from the last theorem m Chapter 3, 1 e , we claim that under (*) * + V -* (* + y) (M) = i (A/) + V(M) , if a is a scalar aX-> {ca){M) = ax(M) and W -* (xy) (Jlf ) = x(M)y{M) Having realized that (•) is a homomorphism one might next inquire about the kernel of (•) If x is a member of the kernel then x(M) = 0 for all M £ M This is equivalent to saying that a: is a member of every M £ M, or that x belongs to the radical of X If the kernel consists only of the zero element however then the mapping (*) must be 1-1 or is actually an isomorphism We can summarize these results by the following Theorem If X is senusimple then X F[St) (•) x — » x(A/) is an isomorphism Before proceeding further a concrete example of these results will be given Example As commutative Banach algebra with identity take w = Jx(t) = £ ‘W" I i, I I < °°| where 11*11 - SW-1 (See Example 3 m Chapter 2 for some discussion of W ) 4. Gel'fand Theory and Point Set Topology 59 First we would like to determine what the class of maximal ideals is in W. Let xo = e xl and let xo(M ) = a. Now consider xj4 = e~ il . By the above-mentioned homomorphism it follows that Xq 1 (M) — or 1 . Using part (7) of the last theorem proved in Chapter 3 we now have the following two equations: f«l = I MM) | < || x„ || = 1 I a" 1 1 = I M(M) | < H I) = 1 which implies M = i. Since | a | = 1, then there must exist a U £ £0, 2v) such that cc = e*‘°. At this point let us consider the mapping (») for W: iso g: W —* W/M — > C x > x(M) iint y £171 fft and ultimately CO CO x(<) = Ecne- - ]C c " ei "'° = x(M). (1) — CD — 03 Let us examine this last statement in more detail. Cer- tainly, since g is a homomorphism, for any finite sum, Sn £ W, Sn = 2-tfC„e <n( we have Sn(M) = >jLvc„e''"°. But we also have | x(M) | < || x (| in general which, in this case, reduces to | S n (M) - x{M) | < || Sn - x ||. Since the last term on the right must go to zero as N becomes infinite, we have established continuity of g in this case which justifies (1) 4 Thus, to each maximal ideal M f Actually the mapping on p. 51, of which g is a special case, is always continuous by part (7) of the theorem on p. 52. 60 Elements of Abstract Harmonic Analysis is associated a point t a € f0, 2i r) such that M consists of those x € W which vanish at to (see part (5) of the theorem on p 52) Conversely, let U € C0> 2») and consider the set m =. e x | £«•■» - o) It is clear that if is an ideal m IV We would now like to show that it is maximal Suppose I was an ideal in W such that 7 3 if properly Then there must be a y £ I such that y <£ M Since y $ M, y(to) ^ 0 Now for any 2 6 W we can write 1(0 = m vW + H " iw» <0 ] But the first member on the right is in I w hile the second vanishes at f<> and is therefore in M This implies that z € I But z was any element of W hence I = IF Let us now review what we have done First we saw that given any if there corresponded a U € fO 2ir) Now we have that given any t 0 € f0 2r ) we can construct a maximal ideal Hence there exists a one to-one correspondence between the set if of maximal ideals of }V and the real numbers in the interval fO 2 it) or M <-* [0 2jt) Consider an x(t) = S-» Cb€ "" such that it is nonvamshmg for every t This implies x $ if for any if € if Tins says that x(t) is a unit or that 1/a: ( <) £ W This result was first proved by Wiener in an entirely different context and is stated below Theorem (Wiener) If is absolutely convergent and vanishes nowhere, then (l/V.^^e'" 1 ) can be expanded in an absolutely convergent trigonometric series Topology This discussion is intended only as a brief introduction, for more detailed information the reader is referred to the books by Bourbaki flj or Kelley [2] 4 . Gel'fand Theory and Point Set Topology 61 A Topological Space Let X be a set and let 0 be a collection of subsets of X. If the collection of subsets satisfies (1) U a O a € 0, a € A, where each 0„ € 6 and A is an arbitrary indexing set, ( 2 ) n" =1 o,- e o, n < », o, e 6, (3) X, 0 € 0, then the pair (X, 0) is called a topological space. Usually the members of 0 are called open sets, and 0 is called a topology for X. Frequently one speaks of just X as being a topological space. Examples of Topological Spaces Example 1. X an arbitrary set and 0 = (0, X}. This is called the trivial topology. It is easy to verify that this is indeed a topological space. Example 2. X an arbitrary set and 0 = P(X) where P(X) is the collection of all subsets of' X and is usually called the power set. This is usually called the discrete topology because every subset, even discrete points, are open sets in this topology. Example 3. Let X be euclidean n-space and let 0 be the collection of open sets as they are usually defined in euclidean n-space. Example 4. Let X be an infinite set and let 0 consist of 0, X, and any subset A whose complement CA is finite. We will actually verify that this is a topological space. (1) C( = D a C(A a ), where A a 6 6 , which is clearly finite. Hence 6 is closed with respect to arbitrary unions. (2) Let A h Ag, •••, An € 6. The intersection' (fLiU, 6 6 , since C( rfLiXi) = U^UCd.,-, is finite.^ (3) This requirement is satisfied by the way 0 was defined. 62 Elements of Abstract Harmonic Analysis Further Topological Notions In the following definitions ( X , 0) is assumed to be a topological space Definition 1. A set F is said to be closed if the complement of F is open, i e , CF € 6 Definition 2. An element x € X is called an adherence point of a subset E C X if every open set containing x contains a point of E Remark It immediately follows from Definition 2 that every element of £ is an adherence point of E Definition 3. The set of all adherence points of a subset E, denoted by L, is called the closure of E By virtue of the remark following Definition 2 we have E <Z E Definition 4 Let X and Y be two topological spaces The function / X — » 1 is said to be continuous if for any open set, 0, in r, / ‘(0) is an open set in A r Definition 5. Let X and Y be two topological spaces The function / X — * Y is said to be a homeomorphtsm if (1) the mapping is both 1-1 and onto, and (2) / is continuous and / 1 is continuous or, to summarize the second requirement, / is bicontinvovs We note that a homeomorphism, since it preserves open sets, preserves all topological properties, abstractly the two sets are indistinguishable topologically We now proceed to the intuitively plausible result that the composite function of two continuous functions is itself continuous Theorem 1. Let X, Y, and Z be topological spaces and let the functions / and g be continuous functions X-> Y —* Z 4. Gel'fond Theory and Point Set Topology 63 Then the composite function h(x) = g(f(x)) is also continuous. Proof. Let 0 be an open set in Z. Then h- l (0) =/- I (r‘(0)). Since g is continuous, g~ l (0) is an open set in Y. Since / is continuous, f~ 1 (g~ 1 (0)) is an open set in X which completes the proof. Before proceeding to our next result we state, without proof, the following: Theorem 2. If A C B, then A C B. The proof of this statement is quite short and the reader should verify it for himself. Theorem 3. Let ( X , 0) b e a topological space and let E\, E? C X. Then Ei U IE — E, u La- Proof. Certainly Ei C Ei l) Ez which implies, by Theorem 2, K\ C U $2- (2) By the same reasoning, we also have E 2 CZ El U E 2 . (3) Combining (2) and (3) we obtain Ki U Kn C Ei U Ez. Now let x be any element in E\ U £•>. This means, by the definition of closure, that any open set, 0, containing x must also contain a point of Ei U Ei. Now there are two possibilities: either (1) every open set containing x contains points of E\ which would imply x £ £\ which means x £ Ei U Ei, or (2) there exists an open set Oi containing x such that Oi n Ei = 0. Suppose this is the case and let 0 2 be any open set containing x. Now we have x £ 0 ifl 0 : . 64 Element] of Abstract Harmon c Analysis Since Oi fi 02 is an open set containing x we must have (0! fl Ot) fl (£, U Ei) ^ 0 But since 0i n Ei => 0 we must have (Oi n 0 2 ) n E, ?f 0 which implies Oi fl Ei 5 ^ 0 or x € £j and hence a € £i u £, Theorem 4 2 =■ £ Proof By the remark following Definition 3 we have £CE Thus all we need show is that Ic£ To this end fet x £ 2? and let 0 be any open set containing x Thug O n £ * 0 or there must exist ay 6 0 such that y 6 £ Since y 6 £ and y £ 0 0 fl £ ^ 0 Thus any open set containing a (: £ must also contain points of E Hence a € £ or 2? C £ Combining this with £ C 2? we obtain £ - Z? QED The next theorem will give us a more useful analytic way of describing a closed set Theorem 5 £ is closed if and only if E = £ 4. Gel'fand Theory and Point Set Topology 65 Proof ( Necessity ) . Suppose E is closed. By definition, CE is open. Let x € CE. Since CE ft E = 0, x $ E. Since no point of E can be in CE, we must have E C E. But, in general, E C E. Hence E = E. ( Sufficiency ). Suppose E = £ and let x € CE. Certainly x cannot belong to E because E = E. Hence there must be at least one open set 0 containing x that has nothing in common with E or, for any x in CE, there must exist an open set 0(x) such that 0(x) fl E = 0. Thus CE C U 0{x). xzCE Since no 0(x) has anything in common with E we have CE = U 0{x). xcCE Since we are able to write CE as a union of open sets, CE is open. Therefore E is closed. QED The next theorem yields another very important property of continuous functions; namely, that they map closures into closures. Theorem 6. Let X and Y be two topological spaces and let /: X -f Y. Then f is continuous if and only if for every set E C X nE) cm. Proof ( Necessity ) . Suppose/is continuous, and let E be any subset of X. Let x 6 E. Now take any open set 0 in Y con- taining f(x). f(x) € 0 implies x 6 / -1 (0) which must be an open set since / is continuous. Since x € E, however, / _1 (0) fl E 0 which impl ies 0 fl f(E) ^ 0. Therefore f(x) 6 f(E) or f(E) C f(E), which proves that the condi- tion is necessary. ( Sufficiency ). Suppose the condition is satisfied and let 0 2 be open in F. Let U = / -I (0 2 ). 66 Elements of Abstract Harmonic Analysi We next note that f(CU) — f(X) n COt The condition implies f(CU) C /(A) n COi Let x € CU and suppose x€Ulix£U~ then f(x) € Oj But we also have f(x) € J(X) f) CO, which implies Ot n (/(X) n C0 2 ) * 0 which is clearly impossible because 0 2 n COj => 0 Hence x $ V orx € CU which impl ies C U C CU Since CU C CU, m general, w e have CU = CU which, by Theorem 5, im- plies CU is closed Therefore U is open which demonstrates that the condition is sufficient QED We now proceed to an alternative way of topologmng a space, called The Neighborhood Approach Let X be a set and let x € X Then the collection of sub- sets of X, denoted by V(x), are called neighborhoods of x, if they satisfy the following four conditions 1 Suppose U € V(x) Then if N D U, N e V(x) In words, this states that if U is a neighborhood of x, then all supersets of U must also be neighborhoods of x 2 If U € V(x), then x £ U 3 V (x) is closed with respect to finite intersections, le, if U lt U t , --•,£/„£ r(z), then P^U, € V(x) 4 Let V g F(x) Then there must exist a set If € V(x) such that for all y m 17, V 6 V{y) Loosely phrased this says that if I' is a neighborhood of z, then it is also a neigh- borhood of all points "sufficiently close" to x Having defined these neighborhoods we can now define an "open" set in terms of these neighborhoods The use of the word "open” will be justified by showing that the “open sets so defined do indeed satisfy the three axioms for a topological space 4. Gel'fand Theory and Point Set Topology 67 Definition 6. A set 0 C X is called open if for any x £ 0 there exists an N (x) 6 V (x) such that x £ N(x') C 0. To paraphrase this one might say that a set is open if it is a neighborhood of each of its points. We will now show that these open sets do give a topology for X. Clearly X and 0 are open according to this definition. 1. Arbitrary uniom. Denote by 0 a (where a ranges through some index set A ) the collection of open sets defined by the above definition and let w = uo„ a where a ranges through B C A. If x £ W, then, for some a, x 6 0*. This implies there exists an N(x) £ V (x) such that x € N(x) C O a C W. Therefore W is open according to this definition. 2. Finite intersections. Closure will be demonstrated for an intersection of two open sets from which it immediately follows, by induction, for any finite number. Let Oi, 0 2 . be open sets, and let x £ Ox n 0 2 : x £ Ox => x € Ni(x) C 0 1 x £ 0 2 =» x 6 lV 2 (x) C 0 2 . Hence x 6 (JVi(s) n N 2 (x)) C (Oi n 0 2 ). By property 3 of neighborhoods it follows that Ni{x) PI IV, (i) is a neighborhood of x. Hence Cfi fl 0 2 is an open set. In summary, we have seen the following process take place: Given neighborhoods, V(x), they immediately give rise to a class of sets, 0, that satisfy the axioms for a topo- logical space. Diagrammatically, X, F(x) \ 6 6S Element* of Abttracf Harmonic Analysis We -will now show that this picture can be extended We will now start with a topological space and define a class of sets V (re) from the open sets that will be shown to satisfy the neighborhood axioms This will give us the following diagram X, V(x) N 0 \ V(x) It will turn out that the classes of sets V (x) and f (x) are actually the same Definition 7. Let ( X , 6) be a topological space U will be called a neighborhood of x if there exists an open set 0 £ 0 such that x £ 0 and 0 C U Loosely phrased this says that a neighborhood of x must be "big enough to fit an open set about the point x The set of neighborhoods of x will be denoted by ^(x) first it will be shown that the class of sets defined by Definition 7 satisfy the four neighborhood axioms Axiom i Let U £ V'(x) and suppose V D U Since U £ V(x) there exists an open set 0 £ 6, such that x £ 0 and U 0 which implies V 3 0 Axiom 2 It is clear from Definition 7 that if V £ ^(#)> then x £ V Axiom 3 Let V» and W be neighborhoods of x Then we have open sets, Oj Oj £ 0 such that x £ 0. C Vi, x € OtCVt which implies x £ O x (\0,CV ifl V t Thus Vj n 1 j is a neighborhood and we have established that the neighborhoods defined by Definition 7 are closed with respect to finite intersections ■4. Gel'fand Theory and Point Set Topology 69 Axiom A:. Let V £ V ( x ) . Then there must exist an open set Ox 6 0 such that x £ 0 X and 0 X C V. But an open set is a neighborhood of any of its points by Definition 7 for, given any z £ 0, where 0 6 0, certainly 26 0 C 0. Hence given any y £ 0 X y £ 0 X c V which implies F 6 V(y). QED We now wish to show that the following diagram is valid: where the arrows pointing down should be read as “gives rise to,” and the arrow going up means that V (x) = V(x). Suppose now that we are given (X, V(x)). Using Defini- tion 6 we can construct a topological space {-X, 0) ; Applying Definition 7 to (X, 6) we can then define ( X , V(x)). We will show V(x) — V (x) . Suppose the above-mentioned process has been performed and let U e V(x). There must exist an open set, O x , such that x 6 O x C U. Since O x is open according to Definition 6 , however, there must exist a neighborhood of x, N(x) 6 V (x) , such that x e N(x) co.cc/. Hence V is a superset of N{x) which implies V 6 V (x) . Thus V(x) C V(x). 70 dements of Abifracf Harmonic Analysis Now let F 6 V(z) and let o- !»n r € vm Certainly x € O and, by the way we have defined 0, Hence y € 0=* F6 V(y) => y 6 F oc f Now we must show that 0 is open according to Definition 6 Let y be any point in 0 As noted before, V 6 V(y) By Axiom 4, pertaining to neighborhoods there must exist IF € V(y) such that for all z 6 TF, V 6 F(z) Hence IF C 0 But y was any point in 0 and we have found a neighborhood of y contained m 0 which implies that 0 is open by Defim tion 6 Thus V € £(x), and combining this with our preuous result, we have (x) = F(x) We now wish to demonstrate the validity of the following situation (with the same inter- pretation of the arrows as in the previous diagram) i e , we wish to show 0 = 0’ Proof Let O 6 6 and let x € O We must find an N(x) € F(ar) such that x € N(x) C O Since 0 6 F(x), we have * € 0 C O which implies 0 6 0' Conversely, let 0 6 0' Then x 6 O implies the existence of N(x) 6 F(x) such that x 6 N{x) C 0 which implies, by Definition 7, that there is a set 0, 6 0 4. Ce l'fand Theory and Point Set Topology 71 such that x6 0 X C N(x) C 0. Hence 0 can be written 0 - U 0 X xeO which imphes 0 £ 0. QED Exercises 1. Let X be a commutative Banach algebra with identity. Let R be the radical of X. Show that X/R is semisimple. 2. Let Xi and X be two topological spaces. Prove that a 1-1 onto mapping /: Xi — > X is a homeomorphism if and only if f(A) = /(A) for all A C X- 3. Construct an example to show that the closure opera- tion is not necessarily preserved by a continuous mapping. 4. Let X be a topological space and let E C X Prove that E is the smallest closed set containing E. 5. Let (X, 6) be a topological space and let E C X. x 6 E is called an interior point of E if there exists an 0 £ 0 such that x 6 0 C E. Let E° denote the set of all interior points of E. Prove the following: (a) (E n F)° = F° n F° (b) CE = ( CE )° (c) C(E° ) = CE (d) (E°)° = E° (e) E a is the largest open set contained in E. CO CO 6. Give an example for which U X ^ U i=l i-l 7. Give examples for which (a) E n F E n F (b) (E U F)° ^ U F° 8. Let X — Z, the set of integers, and let p be a fixed prime. V will be called a neighborhood of n if U contains all 72 Element* of Abstract Harmonic Analys s n + wp* for some k and all m — 0, ±1, ±2, • Show that the neighborhood axioms are satisfied / 9 Let Xi and X 2 be two topological spaces, and X 2 —* X 2 Show that / if continuous if the inverse image of every closed set is closed and conversely 10 Let Xj and X* be two topological spaces and Xi — * Xi Show that / is continuous if for each x £ Xi and any neighborhood Vs of /(x) there exists a neighborhood Ft of x such that /(Ft) C F 2 and conversely References 1 Bourbaki T opologte General* The reader might consult this book for a detailed discussion of the neighborhood approach 2 Kelley General Topology CHAPTER 5 Further Topological Notions In this chapter we will pursue further the topological no- tions introduced in Chapter 4, and terminate the study of topology, for its own sake. Finally we will apply the topo- logical notions to the set of all maximal ideals of a Banach algebra. Bases, Fundamental Systems of Neighborhoods, and Subbases The class of all open sets, 0, in a given space can be a very large class indeed. If possible, given a class of open sets, 0, we would like to cut down the number of sets we must .focus attention on and concentrate on a smaller class of sets. Definition U Let (X, 0) be a given topological space and consider a collection of open sets \B a \. This collection is said to form a basis for the topology if every open set from 0 can be written as a union of sets in { B a ) - Example 1 . Consider the euclidean plane with the usual open sets as the topological space. The class of all open circles, open rectangles, open crescents, etc., are all bases for the topology. Definition 2. A collection of open sets, {B a }, in a topological space, (A, 0) , is said to form a basis at the point x if for every open set containing x there is some member of {B a j con- taining x and contained in the open set. Symbolically, if x 6 0 where 0 is an open set, there must exist a set B x G \B a ] such that x 6 B z C 0. Definition 3. A topological space is said to satisfy the first axiom of countability if there exists a countable basis at every point. 73 N 74 Hemenft of Abttraci Hormone Anolyi t Example 2 Suppose X is a metric space with the family of spherical neighborhoods as the (neighborhood) topology At any point of the space then it is readily seen that the class of spherical neighborhoods with x as the center and rational radii forms a countable basis at the point Thus any metric space satisfies the first axiom of countability Definition 4 A topological space is said to satisfy the second axiom of countabiltty if there exists a countable basis for the topology Definition 5 Let ( X 1 (x)) be the topological space where the class of sets 1 (i) are neighborhoods A collection of neighborhoods B(x) of x is called & fundamental system of neighborhoods of xd for each neighborhood of x N{x) £ V (x) there exists a neighborhood B € B(x) such that B Q N(z) Theorem I Let X be an arbitrary set and consider a class of subsets of X B(x) such that at any point x 6 X the following three axioms are satisfied (BI) If V t Vi 6 B(x) then there must exist a set Vt € B{x) such that C Vi 0 V t (B2) If V £ B(x) there exists a set V € B( x) such that t C V and if y £ V there must exist a member of B(y) contained in V (B3) Each member of B(x ) contains x Then there is a topology for X such that at any point x € X the class of sets B(x) forms a fundamental system of neighborhoods of x Proof We define a class of sets V (x) as follows A set AT £ V(x) if and only if there is a set B £ B{x) such that B C. N We will now prove that the class V(x) forms a neighborhood topology for A Hence we must show that the four axioms Cot a collection, of seta to be called neighbor hoods are satisfied (see p 66) 5. Further Topological Notions 75 1. Supersets. Suppose N 6 F(x) . This implies that there exists a V such that N D V € B{x). Certainly any set IF that contains N also contains V. Therefore W 6 V (x) . 2. The element must be in all its neighborhoods. Suppose N 6 F(x). Then N D F € B(x). By B3 though, x £ F, which implies x € Ah 3. Closed with respect to finite intersections. It suffices to demonstrate that the intersection of any two sets in F(x) is also in F(x). To this end let N\, Ah 6 F(x). Then 2Vi D Fj € B(x) and jV 2 D F, <= B(x) which implies Ah n Ah D Fi n F 2 which, by Bl, must contain a set F 3 £ £(x), or N x n N 2 D Fj 6 B(x). Therefore Ni n Ah € F(x). 4. A neighborhood of x is a neighborhood of all points “sufficiently close” to x. Note that as yet we have not made use of B2.. It-is here that we shall need it. Suppose N 6 F(x) . This implies there exists a F such that A T 3 F 6 B(x). By B2 we have a set F' such that N D F D V e B(x) and if y 6 V', there must be a set B £ B(y) such that B C V' C N. Hence V' € V(y) and for all y € F', N € V(y). This completes the proof. We would now like to focus attention on an even smaller class of sets than the sets in a basis for a topology. Definition 6. A collection of open sets in a topological space is said to form a subbase ( subbasis ) for the topology if the collection of all finite intersections of sets from this class forms a basis for the topology. Example 3. Consider the real line, R, with the open intervals as the class of basis ^ets; i.e., all sets of the form (a, 6) . A subbase for this topology would be the class of all sets of the form (a, « ) , (—«,&). 76 Elements of Abstract Harmonic Analyst Theorem 2. Let X be any set and let S be any collection of subsets of X Then there exists a topology for X in which S is a subbase Proof To simplify matters we mtroduce the following notation B the class of all finite intersections of sets in S O the class of all unions of sets m B 1 fl, X 6 6 To assure this we adopt the following con- vention, vacuous unions represent the null set and vacuous intersections represent the entire space t If the reader finds this convention distasteful, the alternative is to just put the sets 0 and A in 6 separately 2 Arbitrary unions It is clear that arbitrary unions must be m 6 by the very definition of 6 3 Finite intersections We will show that the intersection of any two sets in 6 is also m 6, the extension to any finite number then follows immediately by induction First it will be shown that B is closed with respect to finite intersections Let Bi Bs € B Then 5i = (\S , Bt = ns . where each of the S, S, 6 S, Bi n B, = ( ns *) n ( nSl ) = Si n n S, n 5i n • n St ^ B t To render this intuitively plausible one notes that taking the UD, ®“ of fewer and fewer seta leads to fewer and fewer points being contain in that union whereas taking the intersection of decreasing numbers o sets leads to more and more points being contained in the intersection 5 . Further Topological Notions 77 Now let 0 = Oi fl O 2 where 0 1; 0 2 € 6. Then Oi n Oi where {B al \, {B a ,\ are families of sets in B. Hence 0i n 0 2 = U U(B ai n B„ 2 ) . ttl 02 But since B is closed with respect- to finite intersections, Oi fl 02 has been written as a union of sets in B. We shall often refer to this topology for X, given S, as the topology generated, by S in a way analogous to the way one defines the space spanned by a set of vectors in a vector space. Just as in the case of the space spanned by a set of vectors, that this space is the smallest subspace containing the set of vectors, and that this subspace was actually equal to the intersection of all subspaces containing the set, we note that it is clear that the topology generated by S is equal to the intersection of all topologies containing S. Definition 7. Consider a set X and two topologies of open sets for X, 0\, and 0 2 . [The spaces (X, 0i) and (X, 0 2 ) represent two possibly different topological spaces.] 0i is said to be weaker than 0 2 or 0 2 is stronger than Oi, denoted by Ox < O 2 , if Oi C O 2 . Suppose X is any set and the collection {X a } is a family of topological spaces and j f a ] is a class of functions such that f a : X — > X a . We wish to assign a topology to X in such a way that each of the/„ is continuous. We note certainly the discrete topology assigned to X will do the job, albeit a rather crude one. A somewhat more refined choice consists of taking the intersection of all those topol- ogies for X with respect to which all the /„ are continuous. This topology is called the weak topology associated with the f a and can also be arrived at by taking the class of sets ( fa* 1 (Of ) } , where O a is an open subset of X a as a subbase, S; i.e., take the topology generated by S. 76 Element* of Abstract Harmonic Analysis The Relative Topology and Product Spaces Definition 8, Let (X, 0) be a topological space and let S be any subset of X The set S together with the collection of sets {0 fl £ | 0 6 0} is said to be the subspace (relahic, induced) topology for S Of course it remains to verify that the collection of sets {£ n O j O € 0} does indeed satisfy the axioms for a topo- logical space, and we shall prove this now 1 Since 0 =* 0 n S and S «» £ n X, it is clear that S, 0 are in the above-mentioned class 2 Since (Oi n S) n (0, nS) = (Oi n o,) n S, closure with respect to finite intersections follows 3 Finally, since (1(0. n S) - (^UO.'jnS, it follows that an arbitrary union of sets of the above class is also just the intersection of an open set from 6 and S which proves that (S, {£ n 0 1 0 € 6}) is a topological space and justifies Definition 8 Next we will consider the cartesian product of two topo- logical spaces and will assign a topology to this space The approach used will work for the cartesian product of any finite number of topological spaces Let (Xi, Oi) and (X*, (5 t ) be two topological spaces and consider Xj X X* = { (ij, x,) | Xi 6 Xi, x* 6 X*J We wish to topologixe this space, hence we must propose some definition of open set in the product space A set U C Xt X Xj will be called an open set in the product space if and only if for any y € V there exists a set 0i X 0i such that y € Oj X Oi C 0 where 0i 6 Oi and 0i 6 Ot It now remains to \enfy that the collection of sets so defined does indeed give a topology to the product space 5. Further Topological Notions 79 1. It is clear that 0 and Xi X X 2 are open according to the above definition. 2. Finite intersections. Suppose U and V are open in the product space. Then if y £ U n V, we have y € U => y € Oi X 0 2 C U (a) and y 6 V => y e 0[ X 0; C F. (b) Combining (a) and (b) we obtain y £ (OiX 0 2 ) n (0[ x 00 C n F or ye (Ox n 00 x (0 2 n 0 2 ) C O n 7. Hence 17 n V is open according to the above definition. 3. Arbitrary Unions. Suppose { } is a collection of open sets in the product space. Then if y e U a U a , there must exist an a such that y e U a . This implies y e 0 X X 0 2 C U a C U U a which completes the proof. Note: Henceforth all subsets and cartesian products, when considered as topological spaces, will be assumed to be topologized by the above topologies. Separation Axioms and Compactness To motivate the ensuing discussion consider a topological space with the trivial topology assigned to it. In a general topological space we will say a sequence [x„] converges to a; if every open set containing x contains almost all of the x„. If the trivial topology has been assigned, how- ever, then we see that any sequence at all converges to any and every point in the space. Thus the notion of convergence in this space becomes totally uninteresting because the limit 80 Elements of Abstract Hormowc Analysis is so completely ambiguous The difficulty with the above problem is that there is not enough “separation” between the poults of the space To avoid this we shall further refine the class of all topological spaces by defining some separation axioms Axiom T t. Given a topological space where any two points, x and y, have the property that there exist open sets, 0, and 0 V , where x € O x and y 6 0„, such that 0* n O t = 0, we shall say the space satisfies Axiom T 2 A synonym for this is saying that the space ts Hausdorff Thus given any two points m a T* space we can separate them by enclosing them in nonmtersecting open sets Also, in a Tj space the limit of a sequence ji»J is unique if it exists Example 4 As an example of a space that is not Hausdorff consider the following X is any infinite set and the open sets are the null set, X, and all those sets whose complement is finite (In Chapter 4, Example 4, it is verified that this actually rs a topological space ) In this space it is impos- sible to obtain any two nonmtersecting open sets, hence this cannot be a Hausdorff space One can, however, given any two points x and y, find an open set containing x and not containing y Clearly the open set X — {t/i will do the job Hence, even though we do not have the strong separation imposed by the Hausdorff axiom, we do have some separation, and spaces which do obey this law of separation are classified by the following Axiom T t . If, given any two distinct points of a given topological space x and y, one can find an open set con- taining x but not y and also an open set containing y but not x, then the space is said to satisfy Axiom Ti Example 5 The following is an example of a space that is not Ti Let X = [0, 1) and let 6 - {0, [0, o) where 0 < a < 1 ) Without verifying that this is a topological space we note that given two points x and y where y > x we can find an open set containing x but not y but cannot find an 5. Further Topological Notions 81 open set containing y that does not contain x, which brings us to our next separation axiom; namely: Axiom T 0 . A topological space with the property that for any two distinct points there exists an open set containing one but not the other is said to satisfy axiom T 0 . The following theorem reveals an equivalent way of saying that a space is Ti. Theorem 3. A topological space, X, is Ti if and only if every one point set is closed. Proof (Necessity) . Let X satisfy Axiom Ti, and consider two distinct points x and y. Certainly y 6 X — (x). Since X is Ti though, there exists an open set, 0„, such that y £ 0 V and x $ 0„. Now consider U O v = X — {xj. ycX—lzl Hence X — {xj is an open set and the one point set (x) is therefore closed. (Sufficiency). Suppose for every x £ X that (xj = {x}. If x 5^ y, then y £ X — {x} which is an open set and x £ X — [y\ which is also open. Thus the space is Ti. Definition 9. Let A be an indexing set and let (X, 0) be a topological space. Further, suppose {(?„), where a runs through A, has the property that X C U a <7 a . Then {<?„} is said to be a covering of X. If the set A is finite, then it is said to be a finite covering of X. If each G a is open, the cover- ing is said to be an open covering of X. Definition 10. A topological space is said to be compact if, from every open covering, one can select a finite subcovering. We note that the Heine-Borel theorem for the real line is just the statement that closed and bounded sets there, with respect to the basis consisting of open intervals, are the compact sets. 82 Elements of Abstract Harmonic Anolysls Theorem 4 A compact subspace of a Hausdorft space » closed tively, such that 0, »». ' s) Slnce s P'^^ t -°UU?o"Xc^ T«.e m »^nS 0 ,J such that & - n ; m y n g v,as not rr ,1 u* rr — n ,11, , Suppose now that r ' |0 mil This would m«n tint V n (0„ n B) ^ ® Lire E 7 , 0 (0„ n Q * 0 which > »«"£ cause of the way the U, were chosen m the firstpta we have S C S C S whrch implies S - « Theorem 5 A closed subspace of a compact space is «®P«‘ Proof Let F be a closed subspace and considerj-he ing covering for F Let F - U 0 L„ where * , ope a the 0. are open Hence the entire spa “ * Lust cover X covering 10.1 u CF A finite number of these m Q _ because X is compact denote these by i Since they cover X we must also have F = U(O.nf) which completes the proof Theorem 6 The continuous image of a ,nd compact Symbolically if / is continuous A, is comp x, is a topological space then if / Xi-*X t j(Xi) is compact Proof Consider any open covering of /(A'i) ” j J This implies X, = Since X , , is compact » ^ a fimte number of these must also do the job, deno by Ui, U t , U Thus Z x = U /-(f/,) =»/(Zx) =/( u /-’(f/,)) C Ut/,, Theorem 7. Let Xi be a compact topological space and X 2 be a Hausdorff space and let / be a continuous 1-1 onto mapping of X\ onto X 2 . Then / must be a homeomorphism. Proof. Let E be a closed set in X 2 . By Theorem 5, E must be compact, and by Theorem 6 ,f(E) must also be compact. By Theorem 4, f(E) must be closed. Hence closed sets must map into closed sets under/, and since it is already 1-1, onto, and continuous, we have the desired homeomorphism. QED When one works with sets there is always a certain duality present; i.e., for every statement about open sets there is a corresponding statement about closed sets. Compactness is no exception, for if S is a compact space we have n U O a = S => there exists U O ai — S a i=l and the corresponding dual statement is n nC„ = 0 => there exists D C„,- = 0 a t= 1 where the C a are closed, which is equivalent to: n if no fl C a> . = 0, then flGL 0 . 1=1 a These ideas are now summarized in the following definition and theorem. Definition 11. A family of sets in a topological space with the property that the intersection of any finite number of them is not null is said to satisfy th e finite intersection properly . Theorem 8. A topological space, X, is compact if and only if for any family, {F a }, of closed sets satisfying the finite intersection property, fl„F a ^ 0. 84 Elements of Abstract Harmonic Analysts Proof ( Necessity ). Suppose X is a compact and let I/*,} be a collection of closed sets satisfying the finite intersection property Suppose non that = 0 => u cf. = x Since CF„ are open sets and X is compact, we have UCF t « X=» C\F X = 0 which contradicts the finite intersection property Therefore ft F„ ^ 0 ( Sufficiency ) Suppose the condition is satisfied and simultaneously that X is not compact If X is not compact, however, there must exist an open covering of X, (0.), with the property that, forever yl, 0. x U 0*, U- • • U 0„ t X which implies CO at D CO., n • • • n C0 ak & 0 Hence the class of sets {CO.} satisfies the finite intersection property and, by hypothesis, this implies ft.CO. s* 0 But the {0„} were a covering so that ft.C 0. *= 0 Hence we have arrived at a contradiction and X must be compact Next we will briefly introduce the Tychonoff topology on a product space and state the Tychonoff theorem The Tychonoff Theorem and Locally Compacf Spaces Consider a collection of topological spaces (X„j where « € A and let O a denote the open sets in X„ IV e define the cartesian product of {X.}, HX a , as all functions x A — 4 UX. such that x(a) € X, A projection mapping is defined as follows pr* tlx. -* X„ x-* *(a) 5. Further Topological Notions 85 A topology for UX a ^ desired in such a way that projection mappings are continuous functions. We state without proof that a basis element for such a topology is Jl0„ such that 0 a € 0 a and almost all (i.e. ail but a finite number) the 0 a = X a . This topology for the product space is called the Tychonoff topology is the weakest topology for which all pr„ are continuous. Theorem 9. ( Tychonoff ). The product space II A*, with respect to the Tychonoff topology, is compact if and only if each of the X a is compact. For proof of this result, the reader is referred to the books of Bourbaki Q] and Kelley [J2]- Definition 12. A topological space, X, is called locally com- pact if for each x £ X there is an open set O t such that x £ 0 X and O z is compact. We note immediately that, for any n, euclidean n-spaee with respect to the usual topology is locally compact. Theorem 10. Let Xi and X 2 be topological spaces such that Xi is compact. Let F be a closed set in Xi X X 2 . Then the projection. of F on X 2 is closed. Proof. Let E be the projection of F on X 2 and let yo £ E. Thus any open set 0 containing yo has the property that 0 n E 0. Consider now the set Go b = \x | (x, y) 6 F, y € Op, y 0 £ 0p\ and note that for any n Goi n Goi n • * • n Go n x 0. To verify that this is so we note that n Go 1 n Ojn ••• n a, C n Go,- >~i For if xc G 0l n 0 , n „. n 0 „ then there must exist a y € 0, n O 2 n- • • n 0 n such that (x, y) € F. It is further noted that Go, n o t n- n o„ cannot be null for any n because yo £ E and 86 Elements of Abstract Harmonic Analysis y o 13 in every Of which implies Oj fl 0* n* • • n 0, n E ?£ 0 Now we can say that the collection of sets {0 o ,} satisfies the finite intersection hypothesis in a compact space Hence there must exist x» € fV?o, 0 which implies! (x p , y 0 ) £ F = F Therefore yt> € E and we have the fact that E is closed QED Theorem 11. Let Xi and X 2 be topological spaces and F and 0 be closed and open sets contained in Ai X A* Let E be a compact space m Xt Then the set U {y | (*, y ) 6 is closed (1) *<£ and the set fi ly I (*. y) € OJ = B is open (2) «r Proof (1) We note that U{y| (x,y) € F\ «K is just the projection of (S X X t ) n F on Xt All we need do now is note that F is compact and apply the preceding theorem (2) Consider U„ £ {y ( (x, y) € C0\ -A By the pre- ceding theorem A must be closed But the complement of A is just B and, hence, B is open Theorem 12. Let X% X 2 and Xt be topological spaces and let / be a continuous function / Xi X X 2 — > X } Let E be a compact set in A'i and 0 } be an open set in A* Then V = € 0% for all (simultaneously) x 6 E] is open Proof Let 17 = ( 0 3 ) Since / is continuous this must be an open set m A'i X Xt Now we can write U - n [y I (x, y) € W openj «r and spply the preceding theorem to conclude that U is ope n f See exercise 7 5. Further Topological Notions 87 A Neighborhood Topology for the Set of Maximal Ideals over a Banach Algebra Suppose X is a commutative Banach algebra with identity. Denote the set of all maximal ideals in X by M . We wish to topologize this set now and we will do this by constructing a class of sets satisfying Axioms Bl, B2, and B3 mentioned at the beginning of this chapter; i.e., we will construct a class of sets which will ultimately form a fundamental system of neighborhoods about any point in M. Let e be an arbitrary positive number and let x u x 2 , • • • , x„ be arbitrary members of X. Now let M 0 6 M. We shall show that the class of sets F(M 0 ; *i, X 2 , x„, «), M£M such that for | x k (M) - x*(Afo) | < € k — 1,2, • • • , n satisfies Axioms Bl, B2, and B3. Since M a itself is certainly in V (Mo] X\, ••*, x n , e), it is clear that B3 is satisfied. With regard to Bl consider the following: Let Vi = F(Af 0 ; xi, x 2 , • • •> x„, «i) V 2 = V (M 0 ', yi, y 2 , • • • , yk, a) ■ It is now clear that Vi nFiDF (Mo) x h • • • , x„, y h • • • , y k , min(e x , e 2 ) ) . We will now show that a much stronger condition than B2 is satisfied. We will show that any member of this class of sets is actually a neighborhood of every one of its points and, hence, in the open set topology associated with the neighborhood topology arising from this fundamental system, the sets in this fundamental system of neighborhoods are actually open sets. 68 Elements of Abstract Harmonic Analysis Suppose Mi € V(M 0 , x u • ■ x„, t) and suppose for every Jc = 3,2, • • ■, n that — xt(M q ) In this case, then, we must have V(M t , x u ••*,*.,«) C V(M 0 , zt, - in, «). On the other hand, suppose that x*(Mj) ^ x*(Af 0 ) for some k In this case consider 6* = t Xk^Mx) ~ x k (M q ) j < c and take 5 = max St * Certainly 5 < « or « — 5 > G Since t — 6 > 0, there must exist a real number e € (0, « — S) Consider V (Mi, x t, i n , c) We shall show that V(Mi, X t , • * *, In, c) C V(M<j, I|, • • ■, Xn, «) Suppose M € V(Mi, ii, •••, i«, c) Then | x*(M) - Xk(Mt) | = | Xt(M) - z k (Mi) + x t (Mi) - i*(il/«) | < | x t (M) - xtiMy) j -f ( x k (Mi) ~ xM) I < c + l <«— 5 + 3 = * for any 1*1, "•,» Therefore Af € y(Af.,u, •••,*„€) Hence given any member of this class of sets we see that it is a “neighborhood” of each of its points and, comparing this to Axiom E2, one sees that this is stronger than what is required Lastly we will show that the space AiT with the topology indicated above is a Hausdorff space, i c , we must show that for any two distinct points there exist disjoint open sets contamm^ the points 5. Further Topological Notions 89 Let the two distinct points be Mi and M 2 . If Mi M 2 , then there exists an a; £ X such that a; (Mi) X x(M 2 ). [See part (6) of the last theorem in Chapter 3-3 Since x(Mi) ^ x(M 2 ) there is some real number e such that | a: (Mi) — x{Mi) 1 > e > 0. We now claim V(M\, x, e/2) ft V(M 2 ; x, e/2) = 0 to complete the proof. Suppose the above inter- section was not null. This would mean that there was some element, M, that was in both of the above sets, or that | x(M) - *(Afi) | < e/2 and | a ;(M) - a ;(M») | < e/2 which, by the triangle inequality, implies | *(Mi) - x(M t ) | < | x(Mi) - x(M) | + | x(M) - x(M t ) | e e < 2 + 2 = £ which is contradictory to the way x was chosen. Hence V(Mi;x, e/2) and V (Mi; x, e/2) are two nonintersecting open sets containing the distinct points Mi and M 2 . Exercises 1. Let Xi and X 2 be topological spaces and Xi X X 2 the product space. Prove that the mappings pr t : Ii X I 2 -> Xi and pr 2 : I ( X I 2 -> X 2 defined by pri(a;i, a: 2 ) = a:i and pr 2 (a: l , x 2 ) = x 2 , where Xi 6 Xi and x 2 £ X 2 , are continuous. 2. Prove that any subspace of a Hausdorff space is also a Hausdorff space. 3. Prove that the topological product of Hausdorff spaces is a Hausdorff space. 90 Elements of Abstract Harmonic Analysis i If X is a Hausdorff space with respect to the topology Oi and if X is compact with respect to the topology Oi and if, further, Oi < 0 ly prove that 61 = 6* / » 5 Let Xi —* Xj where / and g are continuous, Xi a topo- logical space, Xs a Hausdorff space Prove that the set {* | f(x) = g(x ) { is closed 6 Prove that if X is a Hausdorff space and if Ei and Ei are disjoint compact sets in X, then there exist open sets Oi and 0% such that Oi D E\, 0 3 Z) E t and Oi 0 Ot = 0 7 Referring to the proof of Theorem 10, show that (xo l/o) € P References 1 Bourbaki Topologie Gentro.lt 2 Kelley General Topology CHAPTER 6 Compactness of the Space of Maxima! Ideals over a Banach Algebra; an Introduction to Topological Groups and Star Algebras In this chapter we will apply just about everything covered so far about Banach algebras and topology to proving that the space of maximal ideals of a Banach algebra, com- mutative with identity, is compact with respect to the topology assigned to it in Chapter 5. After this, the notion of a topological group and several properties of topological groups will be discussed. Theorem. Let A" be a commutative Banach algebra with identity, e, and let M be the space of all maximal ideals in X. Then, with respect to the topology for M defined in Chapter 5, M is compact. Proof. The line of reasoning that will be employed here consists of making statements such as “if we can prove M has some property, then the proof that M is compact will follow”; i.e., the problem will be successively reduced to different problems which, if they can be proven, will yield the proof. Let x be any element of X and consider the closed circle in the complex plane of radius || x j| which we will denote by S x . By the Heine-Borel theorem for the plane, <S X is com- pact for every x. By the Tyehonoff theorem (Theorem 9, Chapter 5) , the product space, n&. where x ranges through X, is compact. Denote the product space, XI S x , by S. A typical element in S is then the function {ax}iex where a x £ S x . 95 92 Elements of Abstract Harmonic Analysis A basis element in the (Tychonoff) topology for S is then a set of the form IT (a®, Xi, x 2 • • • , t) where o° € S, xi, it, • • • , x» are arbitrary members of X, and t is an arbitrary positive number,consisting of all a € S such that | a„ — ( < * i i < . I «* - «5J < « where aj, is the projection of a® on S xl etc Note that there can be only a finite number of these inequalities because almost all of the projections of any basis set in S must be the whole spare S, Consider now the mapping Si -*■ S /M a fixed element of Af\ M —* [z(M) j*tx V a: ranges through X / Since [by part (7) of the last theorem in Chapter 3] | x(M) \ < || * || we can say x(M) € S , We further note that the abose mapping is 1-1, for given any M\ y* M t there exists an 2 * € X such that xt(Mt) y* xo(Mt) Denoting the images of Mi and Mt by {a, I and 1/3,1, respectively we have y* )3, 0 Thus the mapping is 1-1 Consider the 1 1 onto mapping of M onto its image Si C S l\e now claim that this is a homeomorphism Consider the basis set in Si W (or®, Xi * , x n t) This has as its inverse image those M € M such that j x*(M) — a xi } < t for k = 1, • n Since the mapping is onto St, there must exist an Mu 6 M 6. Compactness of Space of Maximal Ideals 93 such that a 0 = \x(M<i) } xeA - which implies a% = x k (M 0 ) ■ The inverse image can now be written, using the notation introduced in Chapter 5, as V (M 0 ; %b x n , e) which is a fundamental neighborhood in M.\ Hence the mapping is continuous It is clear that the mapping takes basis elements in M into basis elements in Si. Hence we have established the desired homeomorphism. We now make the following: Contention. Si is closed. Before proceeding to the proof of this contention let us note what immediately follows from it, assuming it to be true: Since S is compact, if Si is closed, then Si must also be compact. (A closed subset of a compact space is compact.) If Si is compact, then, since M is homeomorphie to Si, M must be compact also. In view of this let us pursue proving that Si is closed. To prove this we must show that any adherence point of Si must actually belong to Si. Let a 0 = {a°} 6 Si. To show a 0 £ Si we must show that there exists an Mo such that al = x(Mo) for all x £ X. This then will imply that a 0 is an image point under the mapping M —> S and, by the definition of Si, will mean that a 0 belongs to Si. Before proceeding, three theorems, one from linear algebra and two from modern algebra, will be stated. Theorem 1. A nontrivial (not identically zero) linear func- tional is always an onto mapping of the vector space onto the underlying field. Theorem 2 ( Canonical theorem on homomorphism ). Let X and 7 be algebras and let h be a homomorphism mapping X onto onto 7; h: X — > 7. Call the kernel of the homomorphism K. t Showing that basis elements in <S'i have, as their inverse image, basis elements in M is certainly a sufficient condition for continuity. 94 Elements of Abstract Harmonic Analys i Then the mapping shown below from A /A onto Y is an isomorphism Y — » X/K Y *-* (x + K) —h( i) Theorem 3 Let AT be a commutative algebra with identity and let Af be an ideal in X Then X/M is a field if and only if M is a maximal ideal Proceeding with our proof again consider the following mapping Y^C (1) It is claimed that this is a homomorphism and also that the mapping is onto and not identically 2 ero i e we claim the following There is some x € A for w hieh «!^o and if |5 € C then for any x cr®, = pal aif» = aj -f aj and — ojaj Assummg these claims to be true for the moment we have a nontrivial linear functional mapping Y onto C Hence we can apply Theorem 2 above to establish an isomorphism between X/M v and C where M 0 is the kernel of the original homomorphism (I) le we have 1 — ► AT/V, — * C X — » x Mo —* a» Since C is a field however we can now apply Theorem 3 above to conclude that Vo must be a maximal ideal Further more since under this latter isomorphism e + V* — * I «*{< + A/o) — * aj we must have al(« + A/#) — x + A/« 53 *(V»)e + A/o If a] - x(A/«) * 0 we could multiplj by 6. Compactness of Space of Maximal Ideals 95 its inverse to conclude e £ M 0 which is contradictory. Therefore a x — x(M 0 ) which was the desired result. Now it follows that Si is closed, therefore compact because S is, and finally, since M is the homeomorphic image of a compact set, M must itself be compact. Let us now return to proving the claims made about the mapping (1) . First we will show that it is nontrivial. Suppose that a° ss 0. Since a 0 £ Si though, for any x\, xz, • • • , x n £ X and any e > 0 there must be some element M £ M such that | x k (M) - <4 | = | x k (M) - 0 | = | x t (M) | < <= for fc = 1, 2, •••, n. (2) In particular, this must also be true for the identity element e. Hence (2) implies | e(M) | < e for any e > 0. But this is impossible, for by the last theorem in Chapter 3 we have | e(M) | = 1. Hence the assumption aJsO has led to a contradiction and we can conclude that the mapping (1) is nontrivial. To prove that (1) is a homomorphism is a straightforward, although somewhat arduous, task. Since the proofs are all in the same spirit, only the fact that the mapping preserves products will be proven here, scalar multiplication and addi- tion being left as an exercise for the reader. To this end con- sider the following neighborhood of a?: TF(a°, x, y, xy, e). Since a 0 € Si, there must exist an M £ M such that the following three inequalities hold: | X(M) -al \ <e j y(M) - c£ | < e | (xy) (M) -«“ v | = | x(M)y (M) - <4 I < <• 96 Elements of Abstract Harmonic Analysis Consider now ! < “ I = | at, - + z(M)(y(M) - o£) + aj(l(jlf) - aj) J <!«!,- x(M)y(M) | + | x(M) 1 1 y(M) - | + -„t| < < + « ii * ii + 1 «; i « because | i(Jlf) | < || * )) Now since e > 0 is arbitrary the result follows Theorem. If X is a complex algebra without identity, then X can be extended to an algebra with identity, X Proof Denote by Jt the set of all pairs (a, x) where a 6 C and x d X Define scalar multiplication, addition, and multiplication of elements from X as follows 0(a, x) = (0a, 0x) where 0 € C ( ai , Xi) -f (as, x t ) = (at + «s, + Xs) (ai, *i) («i. x») = (otiaj, aix 2 + a 2 Xi + XiXj) It is easily verified that X, with respect to these operations, is an algebra It is also easy to verify, by just substituting into the last expression, that (1, 0) is an identity for ft, and that the following mapping is an isomorphism X-X x -♦ (0, x ) Loosely speaking, we will actually make an identification between x and (0, x) and wnte fa,*) = a(l,0) -f (0, x) = ac + i where e *= (1, 0) 6. Compactness of Space of Maximal Ideals 97 Hence we represent X as {ae + x |“ c c |) to obtain the required extension. Using the same notation as in the preceding theorem, suppose that X is a normed algebra without identity. If we define || ae + x || = | a | + || x ||, then X becomes a normed algebra also. If, in addition to being a normed algebra, X is a Banach algebra, then X will be a Banach algebra too and this we shall prove. Let {a„e + x n ] be a Cauchy sequence in X; i.e., for any e > 0, there exists an N such that n, m > N implies || a n e + x n — (a m e + x m ) || < e. This is the same as (| — a n | + || x n — x n ||) < « which implies {«*„} is a Cauchy sequence and that {a; n } is a Cauchy sequence. Since C is complete, {«„} must converge to some af C, and, since X is a Banach algebra, { x„ } must converge to some x € X. Hence <x n e + X* y ae -f x. Therefore X is a Banach algebra. Star Algebras Definition. Let X be a complex algebra. X is called an algebra with involution (star algebra, symmetric algebra ) if there exists a mapping X->X x — » x* such that (1) (ax + py)* = ax* + By* (a, B € C) (2) a:** = x (3) ( xy ) * = y*x*. Listed below are some examples of algebras with involution. Example 1. Consider the class of all continuo us fu nctions over the closed interval [a, 6] and take x*(t) = x(t ). 98 Element] of Abstract Harmonic Analysis Example 2 Let X be a Hilbert space and consider L(X, X) , the class of all bounded linear transformations on X As noted previously, this is an algebra Let A € L{X, X) and take A* to be the adjoint operator of A Example 3 Consider W the class of all absolutely con- vergent tngometnc series, i e , ft where £ | c» j < w Let x £ W and take x*(t) =* £ L.e"' Topological Groups Definition A set G is said to be a topological group if and only if G is a topological space and (G1) G is a group (G2) G is a Hausdorff space (G3) the mapping G —> G g — ff -1 is continuous (G4) The mapping (denoting the group operation as multi- plication) G X G — * G (gi ffs) gi?2 is continuous It is to be noted that the linkage between the topological and group structures is provided by Axioms G3 and G4 In the axioms and in the ensuing discussion the prototype group operation wilt be taken to be multiplication and several examples of topological groups are listed below 6. Compactness of Space of Maximal Ideals 99 Examples of Topological Groups Example 1. Let G be any abstract group with the discrete topology assigned to it. Example 2. The real and complex numbers with respect to ordinary addition and the usual topologies. Example 3. R* and C* (the real and complex numbers with zero deleted) as a multiplicative group with the usual ■ topologies assigned. Example 4. M n (R ) : the class of all n X n matrices with real entries, and M„(C ) : the class of all n X n matrices with complex entries. As the group operation we shall take addi- tion of matrices. To topologize the above closses we introduce the following distance function for either case. Let A = (ay) and B — (0y) be matrices with either real or complex entries and take the distance between A and B to be d(A, B) = max | a,-, - /9y j. if) Now, since we have defined a metric on the space, we can take the topology induced by considering the “spherical” neighborhoods as basis elements. Example 5. GL„(R ) : the general linear group of n X n matrices (nonsingular) with real entries, and GL n ( C) : the general linear group of n X n matrices (nonsingular) with complex entries. As the group operation we will take ordinary multiplication of matrices. Noting that GL n {R ) C M„{R) and GL n (C) C M n (C) we shall take the topology to be the induced topology from the bigger space in each case. Example 6. U n (C ): ( the unimodular group) all complex n X n matrices with determinant equal to one. We will take the group operation to be multiplication of matrices and noting that C7„(C) C GL n (C) we will assign the induced topology from <?L„(C) to U n {G). 100 Elements of AbsJroeJ Hormon c Anolyj : Example 7 The class of all complex valued continuous functions on a compact metric space X We shall take the law of composition to be ordinary addition For a topology we assign sup«x {/(*) j as norm to this space and will take the spherical neighborhoods so defined to be the basis ele- ments for an open set topology Several observations that immediately follow from the definition of a topological group will now be listed 1 Continuity of multiplication (Axiom G4) is equivalent to the following condition Let gig* =* g% Then for any neighborhood Vt of g 3 there exist neighborhoods Vj of pi and Vi of g t such that Fil sC Vi where T iV, *= [xy | x € V, y >E F,} 2 Continuity of taking inverses (Axiom G3) is equivalent to the following condition For any neighborhood V of g~ l there exists a neigh borhood V of g such that XJ 1 C F where U ' - (ar-M x € V\ 3 Statements 1 and 2 (simultaneously) are equivalent to Let g 3 = Qiffs 1 and V 3 be a neighborhood of <?j Then there exist neighborhoods of gi and g 3 V\ and \ , such that V\Vt 1 C fa Proof It will now be shown that (1) and (2) imply the above condition Let g 3 = g 3 gi 1 and let 1 3 be a neighborhood of g 3 By (1) there exist neighborhoods I 2 and Ut such that Q\ € Ii fft 1 E Ut and V 3 Ui C li By (2) now since U t 13 a neigh borhood of g t 1 there must be a neighborhood T * of Qi such that 1 f 1 C Ui Hence ViF, 1 C L which completes the proof that the above condition is necessary for (!) and (2) 6. Compactness of Space of Maximal Ideals 101 Sufficiency. We first show that the condition implies (2). Write eg _I = g~ l and let W be a neighborhood of g~ l . The condition implies that there exists a neighborhood, U, of g and a neighborhood, T 7 , of e such that vu-' C W. Since e £ F, U~ l C VU~ l . Hence U~ ! C W which proves that the condition implies (2). Next, we show that the condition implies ( 1 ). Let = g s , or Let V 3 be an arbitrary neighborhood of g 3 . Then, by statement ( 3 ), there exist neighborhoods Fi and Ui of g h g: 1 , respectively, such that TW c Vs. Now since Vi is a neighborhood of g 2 \ by what has already been shown, we have that there exists a neighborhood F 2 of g 2 such that V 2 l C U s , or F 2 C Thus ViVi C C Fa- The following two statements follow directly from the preceding. 4 . Taking the product of any three elements of G is a continuous operation; i.e., let gig 2 g 3 = g* and let F4 be a neighborhood of gt . Then there exist neighborhoods of <71, gi, and r/a, Fx, Vi, and V 3 , such that ViV 2 V 3 C F<. • 5 . Let .<7j (72 = g 3 and let V 3 be a neighborhood of g 3 . Then there exist neighborhoods of gi and Qi, Vi and F 2 , such that V\V 2 c f 3 . These results will now be absorbed into the following observation. 6. Let &g k V---g’n , ‘ = 9 102 Elements of Abstract Harmonic Analysis and V be a neighborhood of g Then there exist neighbor- hoods of g it g 2 , •• •, and g n , V t , V t , • • and V„ such that • v*' c V 7 The following mappings are all homeomorphisms (1) G-+G (2) G~+G (3) (?—►£? 9 -* 9°9 9 -> 99» 9~* r l , where is a fixed element of G Proof Since the proofs of these three statements are all about the same, only (2) n ill be proved here, the proofs of the rest being simple to prove following the same lines of reasoning (2) is 1-1 Suppose g,go = g z go This immediately implies ffi = after multiplying on the right by g 7 l (2) ts onto For any g 6 G it is clear that gg 7 x will map into it under (2) (2) is continuous Let h * ggc, and let V be a neighborhood of h By the very first observation there exist neighborhoods of g and go U and IF such that UW C F which implie* Ugo C V which establishes the continuity of (2) (2) is open (takes open sets into open sets) Since we can write the inverse function explicitly as (2)~ J <7— ♦ G 9~* 997\ th ^rixist neig iis is exactly the same as what was just proved is a ho meomorphism Proof ^ p ^ a c | osec i set m G and let 0 be an open set in above con snJ e i cmmt ol G Then Let pi there ex g«F, Fgc and F~ l are all closed barbo^er d Vts an open set m G and E is any set in G, then that | UE, EU, and U~‘ proo i ire ill open 6. Compactness of Space of Maximal Ideals 103 Proof. We will prove here that Fg<> is closed and that U E is open to illustrate how the proofs run and leave the rest as an exercise for the reader. Fg 0 is closed. Consider/: G —* G where /(g) = gg 0 , g 0 fixed. By 7, / is a homeomorphism. Therefore, since F is closed, /(F) = Fgo is closed. UE is open. Since U is open, /([/) = Ug 0 must also be open. Now we can write UE = U(Uffo); i.e., as a union of open sets. Hence UE is open. 9. Any topological group, G, is homogeneous ; i.e., for any g h go € G there exists a homeomorphism of G onto G that takes gi into g 2 . To prove this all we need consider is the mapping /: (?-><? 9 997*92 and note that, since g/'g 2 is fixed, this mapping, by 7, is a homeomorphism. As to the importance of this result consider the following situations: 9.1. Suppose we wished to show that G is locally compact. We desire for any g £ G an open set containing g such that the closure of the open set is compact. If we can show, just for the identity element, say, that there is an open set con- taining it whose closure is compact, we can apply 9 and con- clude that this is true for every element in the space. 9.2. Suppose we wished to show that G had the discrete topology. If we could show that just the set (e) was open this would, by 9, immediately imply that this was true for every element in the space. 1 0-4 Elementi of Abstract Harmonic Analysts 10 Suppose V is a neighborhood of e Then there exists s neighborhood U of e such that U Q V and V = U~* In words, this states that within any neighborhood of the identity there is a symmetric neighborhood of the identity Proof Let U = V fl V~ l It is now clear that U is a neighborhood of the identity and that U C V It remains for us to show that U is symmetric Before proceeding to that we note that this proof will apply to open sets as well as neighborhoods because if 0 is an open set such that V2)0, K->DO-», then V n D 0 n O- 1 Let a € (! This implies o £ V and a 6 V~ l , but a € V a -1 € F" J and a 6 F -1 =* a~ l € F Hence a" 1 € F ft F' 1 = U In words, then, what we have shown is that if a is m V, then a -1 must be m U also, or that V = U~ l 11 Let V be a neighborhood of e Then there exists a neighborhood of e, U, such that U* C V Proof Let V be a neighborhood of e and write ee = e Now there must exist two neighborhoods of e, and F* such that 1 1 Fj C F Let V — F t H Fj Now we have U* = UU C Fit * C F This result can easily be generalized by induction, to the case where if F is any neighborhood of e and n is any positive integer there must exist a neighborhood, U, of e with the property that {/* C F 6 . Compactness of Space of Maximal Ideals 105 12. Let E be a compact set, 0 an open set, and G a topolog- ical group. Suppose E C 0 C G. Then there exists some neighborhood V of e such that VE C 0. Proof. Let g £ E C 0 and write g = eg. We can find a neighborhood of the identity, U„, for each g such that U 0 g C O. By 11 there exists a neighborhood, which we may assume is open, of e, W„, such that IF* C U 0 . Now we can write EC U W 0 g geE as an open covering for E. Since E is compact we must also have gi such that n EC U W 0i gi. Consider now the neighborhood of identity V = TF 0 I nlF ff! n..- nw, n . It is clear that V C for any j. For any element, g, of E there must be some j such that 9 € w aj g, =* Vg C VW 0i g,- C W C C 0; whence Therefore Vg C 0 for any g 6 E. VE = U Vg C 0. oeE 13. Suppose Ei and Ei are compact subsets of a topological group G. Then the product EiE 2 is compact. Proof. Consider the mapping G XG-+G ( git 92 ) 9i92- 106 Element* of Abstract Harmonic Analysis Since Ei and E> are each compact, the product space E t y E s must also be compact Smce the above mapping is continu- ous, the image of a compact set must be compact Hence E x Et, the image of Ei X E t , is compact Exercises 1 Determine M for the Banach algebra A, of all functions analytic m j z j < 1 and continuous in | z | < 1 2 Suppose 0 is a topology for Af, the set of all maximal ideals of the commutative Banach algebra, with identity, X, such that A/ is compact with respect to O, and all func- tions x (Af) for x £ X are continuous in this topology Then prove that 6 coincides with the weakest topology on iff for which all x(ilf) are continuous 3 Let G be a topological group and let g € G, g ^ < Prove that there exists a neighborhood V of e such that V n gV - 0 4 Let G be a topological group and let g € G If T is any neighborhood of g, show that there exists a neighborhood, V of g such that V C V Topological spaces with this property are called regular References 1 Nat mark Normed Rings 2 Pontrjagin Topological Croups CHAPTER 7 The Quotient Group of a Topological Group and Some Further Topological Notions In this chapter we continue with our study of the ele- mentary properties of topological groups and discuss certain properties of the quotient groups of a topological group. In addition some further topological notions are introduced: in particular Urysohn’s lemma, the Tietze extension theorem, and the One-Point Compaetification Theorem are stated. Locally Compact Topological Groups Definition. A topological group, G, is said to be locally corn-pad if the original topological space, ( G , 6 ) , is a locally compact space. t Definition. The closure of the set of points over which a complex-valued function defined on a topological space does not equal zero is called the support of the function. Theorem 1. Denote by C 0 (G) the class of all continuous, real-valued functions on a topological group, G, with com- pact support (the function must vanish everywhere outside some compact set). If G is a locally compact topological group and / £ C 0 (G) , then / is uniformly continuous; i.e., for any e > 0 there is some neighborhood of the identity, W, such that |/(<?i) — f{gf) I < e for all gig%~ 1 € W. (We note that W may always be assumed symmetric) . Proof. Let / £ C 0 (G) and let E be a compact set outside of which / vanishes. Since G is locally compact there must be some neighborhood Fi of e whose closure is compact; i.e., e £ Vi and Vi is compact. Within any neighborhood of e, however, there must be a 107 1 06 Elements of Abstract Harmonic Analysis Since E\ and £* are each compact, the product space Ei Y E\ must also be compact Since the above mapping is contrnu ous, the image of a compact set must be compact Hence E\E t , the image of Ei X E t , is compact Exercises 1 Determine M for the Banach algebra, A, of all functions analytic in \ z 1 < 1 and continuous in | * | < 1 2 Suppose 0 is a topology for M, the set of all maximal ideals of the commutative Banach algebra with identity, X, such that M is compact with respect to 0, and all func- tions x(Af) for x € X are continuous m this topology Then prove that O coincides with the weakest topology on ifr for which all x(M) are continuous 3 Let 6 be a topological group and let g £ G,gs^e Prove that there exists a neighborhood V of e such that V n gV = p 4 Let G be a topological group and let g € G If V is any neighborhood of g, show that there exists a neighborhood, U of g such that 0 C V Topological spaces with this property are called regular Refebences 1 Naimark Normtd Rings 1 Pontrjagm Topological Groups 7. Quotient Group of a Topological Group 109 which implies f(gg*) — 0, and our claim about inequality (1) has been verified. To fit this into the notation suggested in the statement of the theorem we let gi = 992 for g £ W, or, since g = gig* 1 , gig * 1 £ w. Inequality (1) then states \f(gi) - fig*) I < e for all g wp 6 W. (2) Since W is a neighborhood of the identity, however, it must contain a symmetric neighborhood of identity. This neighborhood, since it is a subset of W, will also imply (2) and the theorem is proved. Remark 1. Note that our definition of uniform continuity involved the statement gig^ 1 £ W. In the general topological group this is not equivalent to requiring gj'g* £ W. In locally compact spaces, however, it is for functions in C 0 ((?). In our particular case if we had taken U' = {?||/(m) ~f(g 2 ) ! < « for all g* 6 VE) we would have arrived at I figf) - fig 2) I < « for some neighborhood of identity, W, such that g^'gi £ W’. It is to be emphasized once again though that local com- pactness is the key feature in being able to make this statement. Subgroups and the Quofienf Groups Definition. H is called a subgroup of the topological group, G, if (1) H is a subgroup of the group (?; i.e., a, b £ H=> ab' 1 £ H, and (2) H is a subspace of the topological space G. We note immediately that H is itself a topological group. no Elements of Abstroct Harmonic Anolynj Definition A subgroup, H, of the topological group G is said to be a dosed subgroup of the topological group if H is a closed subset of the topological space, G Consider now the coset spacef G/R *= {gH\g£G\, and consider the natural or canonical mapping / G — » G/H g-*gH We desire a topology for G/H now that will simultaneously make / continuous and make G/H Hausdorff With regard to satisfying the first requirement consider the following as a possible definition for open sets in G/H W C G/H is open if and only if /-‘(W) is open in G If the class if = {IT C G/H \ is open in G\ truly forms a collection of open sets, then certainly / will be continuous It will now be shown that IF - does indeed meet the required standards 1 Since / -l (0) = 0 and f~ l (G/H) - G, then certainly 0 G/H € W 2 Let | IT a } be a collection of sets in \V and consider /-(urn) Since /-'( u»'.) = u; >( w .) t G/H is a group if and only if H is a normal subgroup of the origins/ group 7. Quotient Group of a Topological Group in and each of the / -1 ( W a ) is open in G, it follows that UWa e W. 3. Lastly, consider TFi, W 2 (L W: f-'iWi n W 2 ) = / -1 (W,) (\f- l (W 2 ). Since / -1 (Wj) and/ -1 (W 2 ) are each open in G, it follows that Wj. n W 2 6 W. Thus we are justified in calling W a collection of open sets, and it now follows that / is continuous with respect to this topology. We will now show that / is also an open mapping (takes open sets into open sets) . Let 0 be an open set in G. We wish to show /(0) is open in G/H ; hence we must show/ -1 (/(0)) is open in G. We first note f(0) = [gH | g € 0] = OH. Suppose now 9i 6 f-Kf(O)) =>/(<h) 6 }{0) = OH which implies there is some g € 0 such that f(gx) = gH or giH = gH or {g\h | h € H] = {gh | h £ H}. Since e 6 H, though, ffi 6 \gih\he H\ = {gh\ h£ H). Hence there must exist an h £ H such that gi = gh which implies gi £ OH or /- ! (/(0)) C OH. 112 Elements of Abstract Harmonic Analysis Conversely, consider any element of OH , gh 6 OH, g € 0, h 6 H Since /(f*> -iM-gHe; f(0), then gt 6 /-(/(0)) Hence /-*(/(0)) = OH which is open in G by observation 8 of the previous chapter Now we would like to show that G/H is Hausdorff, when- ever H is closed Suppose g u gi € G and suppose further giH * g,H, (3) i e gJI and giH are distinct points of G/H It follows from (3) that ffi € g%H [assuming there is an h, £ H such that gi - gh t leads to an immediate contradiction of (3)3 Since H is closed it follows by observation 8 of Chapter 6, that g t H is closed which implies there is some neighborhood U of identity such that Vgi D g t H ~ 0 for there must be some open set, V containing g L such that V 0 g,H « 0 (For example, take V — C(gxH) ) Since V is open, IVT 1 is open and we can take the set U, mentioned above, to be Vgi 1 There must exist also a symmetric open neighborhood IF C G such that IF* C G We now claim WgiH n Wg,H = 0 7. Quotient Group of a Topological Group 113 Suppose not; i.e., suppose there existed w lt w 2 £ W and h, hi £ H such that Wigih = vhg-ihi => wj'wigi = gJhlq 1 . Since W is symmetric, wj 1 £ W ; hence wjwi £ W 2 C U. Therefore Wi'WiQi £ t/^i while ■gzhhi 1 6 g 2 H, which is incompatible with Ug n n g 2 H = 0. Therefore WgJI n WgJI — 0, which implies n Wg 2 = 0 because, since e £ H, Wg 2 C WgJI and Wg 2 C WgJI. Now let W = f(Wg i) = WgJI W” = /(TFflf,) = WgJI. By the above argument W n W" = 0. All we need show now is that gJS £ W’ and gjl £ W" to complete the proof. To this end we note f(gO £ f(Wgi) because e £ W but Hence and, similarly, /to) = gtH. giH £ W' gJB £ W". QED Before proceeding the following definition is needed. 1 1 4 Elements of Abjtnjct Hormotvc Aralyiii Definition. JST is said to be a normal subgroup of a topological group if 1 N is a normal subgroupf of the group, and 2 N is a subspace of the topological space Consider the coset space now G/N where N is a closed normal subgroup of the topological group G We introduce the opera- tion of multiplication to G/N by defining giNgtN = gtfiN and further claim that G/N is a group with this as the law of composition This is a standard result from group theory and will not be proved here By the preceding statement, for any closed subgroup, we know that G/N is a Hausdorff space and would now like to show that, in addition the operations of multiplication and taking inverses are con- tinuous or that G/N is a topological group (Henceforth G/N will be referred to as the quotient group or factor group ) By observation 3 of the preceding chapter we showed that Axioms G3 and G4 were equivalent to ahother condition and it is this condition that we shall show G/N satisfies to con- clude that G/N is a topological group Let IF be a neighborhood of QtNpt'N = g&'N Hence, with / denoting the canonical map as before, gig? € We can now assert the existence of neighborhoods, Vi and F t , of g t and g t , respectively, such that WC /~W =>f(Vi)f{V?) c IF t A normal tubgroup N, of a group G baa the defining property that gNg~' *> A gN *■ Ng for all <7 As will become evident here the normal subgroup in group theory plays a role analogous to that of the ideal to rtng theory 7. Quotient Group of a Topological Group 115 Since Also, 9i 6 Vi, giN 6 /(F ,). fl? 1 € F?‘=* flfW € /(F^)- But, since the canonical map, /, is a homomorphism, we can say W) = /(F 2 )-» or, noting that A must equal A -1 , and Ap 2 = g 2 A, we have <7 2 A € /(F,) which completes the proof. Theorem 2. Let G be a topological group and further suppose G to be compact (locally compact). If A is a closed normal subgroup, then G/N is compact (locally compact) . Proof. First suppose G is compact. Since the mapping f: G-* G/N, f(g) = gN, is continuous and G is assumed compact, we have written G/N as the continuous image (the mapping is clearly onto) of a compact set and we can assert that G/N is compact. Suppose now that G is locally compact. If so there is some open neighborhood of identity, 0, such that the closure of 0 is compact. The image of e, /(c), is just A and thus A = /(e) € f(O) CRO). But, since / is continuous, /(O) is comp act while f(0) is open because / is open; moreover, /(O) C /( 0) because /(0) is a compact subset of a Hausdorff space and, there- fore, closed. Finally /(O) is a closed subset of a compact space and therefore compact. 116 Elements of Abstract Homoroc Anoiyjis Directed Sets and Generalized Sequences Definition. Suppose S is a partially ordered set with respect to < We say S is dtrecied upward if for any x,ymS there exists a 2 € S such that x < z and y < z We also write z > z and 2 > y To paraphrase this one might say that every finite set has an upper bound Similarly one defines a set directed downward Consider any mapping from a directed set S into an arbitrary set X f S — * X / is said to be a generalized sequence in X Suppose that X is a topological space \\ e will say that the generalized sequence, /, comerges to x € X if for any neighborhood, V, of x there exists a t € S such that /(a) € V for all s > t, and will denote this situation by the following notation hm/(a) = x s In a general topological space, x 6 £ does not imply that there is some sequence of elements in E converging to x This will be true, however, in any metric space, or, more generally, m any space that satisfies the first axiom of count&biht} In any case there wilt be some generalized sequence of elements in E converging to x Having intro- duced the notion of a generalized sequence we would now like to introduce a generalized Cauchy sequence Definition. Suppose S is a directed set and X is a metric space and consider / S — » X / is called a generalized Cauchy sequence if for any e > 0 there exists a to £ S suc ^ /(t)) < * for all s, t > to where d is the metric on X Theorem 3 If / is a generalized Cauchy sequence in a com- plete metric space, then / converges, where/ S —* X, S is a directed set and A' is a complete metric space Proof There exists an s« € S such that for h, h ^ t/(fi)) < l/ n where d is the metric on X and n is a positive integer Let r« » sup(si, s t , s,} 7. Quotient Group of a Topological Group 117 We now claim that {/(r n )j is a Cauchy sequence in X. To prove this consider dU(r n ),f(r m )) and suppose n > m. This implies r„ > s m and r m > s m which implies d(/(r„), f(r m )) < 1/m or that { f(r n ) } is a Cauchy sequence. Since X is complete there must exist am£ X such that lim/(r n ) = x. n-+ co Suppose now that e > 0 is given and we then choose an N such that 2/N < e and such that d(f(r N ), x) < e/2. Since s > r N implies s > r N > sn we have d(/(s),x) < d(f(s),f(r N )) +d(/(r w ),x) 6 . e < 2 + 2 "' or lim/(s) = x. s Further Topological Notions We first wish to introduce a type of separation not yet mentioned. Definition. A topological space is said to be normal if it satisfies Axiom T<, where Axiom T 4 is as stated below. 116 Elements of Abstract Harmon c Analys s Axiom T< If Jl and K are disjoint closed sets then there must exist disjoint open sets one containing H and the other containing K Some theorems that we will have occasion to make use of will now be stated and the reader is referred to any book on topology for the proofs Theorem 4 Let X be a topological space and let F and 0 be closed and ojien subsets of X with the property that F C O Then A is normal if and only if there exists an open set V such that F Q U Q U C 0 Theorem 5 ( Urysokns lemma) Let H and K be disjoint closed subsets of a topological space A r Then X is a normal space if and only if there exists a real valued continuous function f such that 1 0 </< 1 and 2 /(*) =* 0 on H and f(x) * 1 on K Theorem 6 (Extension theorem) Let F be a closed subset of a topological space X and let / be a real valued continuous function defined on F Then A is normal if and only if / can be extended to a continuous function on X Theorem 7 ( One point campactifica tt on ) Let X be a topo- logical space Then there exists a compact space X* con taming X and just one other point such that the relatne topology on X induced by X* coincides with the original topology Although we will not prove this theorem here w e will list the essential construction involved in the proof and some facts about it Let j* be anj element not in A (eg lfXisa collection of numbers let x* be the color blue) and consider X* = X U (x*\ and Tl the class of open «ets II in A such that <711 is compact 7, Quotient Group of a Topological Group 119 Define 0* to be an open set in X* if x* ( 0* (=>0* C2), (a) then 0* must be open in X or if a:* G 0*, then 0* n X G W. (b) Under this construction we now contend that the following statements are true (the reader is asked to prove their validity in exercise 6). (1) { 0*} is a topology for X*. (2) X* is compact. (3) The induced topology on X agrees with the original topology. (4) If X is locally compact and Hausdorff, then X* is ’ Hausdorff. Using these results we can now prove the following theorem. Theorem 8. Let £ be a compact set and let U be an open set such that E C U in a locally compact, Hausdorff space, X. Then there exists a continuous real-valued function, /, defined on X , such that 0 < / < 1 and such that f(E) = 1 and f(CU) = 0. Proof. Consider the one point compactification of X, X*, and define Fi — X* — U, F 2 = E C U. We note that Fi fl F 2 = 0. It is now desired to show that Fi and F 2 are closed subsets of X* under the convention established in the preceding theorem and to do this we shall show their complements are open. Since X* — Fi = U and U is open in X, it follows that F\ is closed. Since x* G X* — F 2 we must consider X D (X* — F 2 ) and apply condition (b) of the preceding theorem: Xn (X* - Ft) = X - F 2 = X - E. 120 Element* of Abitroet Harmonfc Analyst* Now, since X — (X — E) == E is compact m X, it follows that Ft is closed in X* By part (4) of the preceding theorem, we have that X* is a compact, Hausdorff space and by exer- cise 8 we have that any compact Hausdorff space is normal Hence, by Urysohn’s lemma, there must exist a real valued, continuous function, /, such that 0 < / < 1 and such that f(F i) - 0 and /(ft) = I Making the required substitutions for Fi and F } and noting that /(X* — U) = {0| implies that /(X — £7) = {0} completes the proof of the theorem Theorem 9. Let X be a normal space and let F be a closed subset of X Let {0,|, t =» 1, 2, • • ■, n, be an open covering of F Then there exists an open covering of F, {(/,), i «= 1, 2, • • •, n, such that V» C 0„ t =■ 1, 2, • • *, n Proof Let A, - CF U^UO.) Then ca , ■= f n (nco,) which implies CAx C O t or CO t C Ai Since X is normal there must exist an open set, V, such that (see Theorem 4) cacvcfc 4, Let Vi = CV We now claim that CA, Ot, 0, covers F, and £?* C Ox To this end consider Ux - C? C CV Vi C CV » CV (CV is closed) But CV C Oi, hence Ox C 0* 7. Quotient Group of a Topological Group 121 Now, since Ai U CA r = X and CUi = V C At =*• CAt C Vi we have which implies and, finally, Ai U lh = X (At OF) UUtD F u Ut D F. Proceeding in the same manner we can construct the other open sets, t/ 2 , ■ ■ • , U n , having the required properties to complete the proof. Definition. Let (/,-}, i = 1, • • n, be a family of nonnega- 'tive, continuous functions defined over a topological space, X, such that = 1» for all x £ X. Such a family is called a continuous ■partition ( decomposition ) of the identity. Theorem 10 . Let {0,-} be a finite open covering of a normal space, X. Then there exists a continuous partition of the identity, {/,}, such that /,( CO, ■) = 0. Proof. By Theorem 9 there exists an open covering, {£/<), of X such that (7,- C 0,- for each i. O, and CO,- now con- stitute disjoint, closed subsets of a normal space and Ury- sohn’s lemma may be applied to assert the existence of continuous functions, {</,}, such that 0 < g, < 1 and and g;(x) = 1 on £7, gAx) = 0 on COi. It is clear that, for every x, J^gAx) > 0 , 122 Elements of Abstract Harmonic Analysis and therefore sensible to consider the functions /.(*) ~ g.(*) 2"-ig.(ar) which satisfy the conditions £/.(*> - 1 (a) /,(*) - 0 if i € CO, (b) and, thus, the theorem Corollary. Let F be a closed subset of a normal space A and let |0,| be a finite open covering of F Then there exist continuous functions | /,} defined on X such that /.(*) > 0 £/.<*) = 1 i € f »-i and /,( i) =0 if x d CO, Proof It is clear that \0,} and CF constitute an open covering of AT By the theorem then there exist nonnegatn e, continuous functions {/.I, g such that /,( i) *= 0 on CO , and g(x) — 0 on F — C(CF ) Further, £ (Jx) + Q(x) = 1 for any x € X, 7. Quotient Group of a Topological Group 123 hence, for x £ F, 33/<(s) = 1 - i=l Exercises 1. Let Gi and G 3 be two topological groups, and f:G l — > G 2 . / is called an iscnnorphism if / is an isomorphism of the abstract group Gi onto G 2 and a homeomorphism of the topological space G 2 onto G 2 . Give an example of two topological groups which are isomorphic as abstract groups but not as topological groups. 2. Let (?i and G 2 be two topological groups, and / : G\ G 2 , a homomorphism of the abstract group Gi into G 2 . Show that f is continuous or open if and only if it is continuous or open at e, the identity of Gi. 3. A sequence of points { x n } of a topological space X is said to converge to the point x £ X (x n — > x) if any open set 0 containing x contains all x n for n sufficiently large. Let E (Z X. x £ X is called a limit point of E if every open set containing x contains a point of E different from x. Give an example of a topological space X and subset E such that a: is a limit point of E, but there exists no sequence in E which converges to x. Prove, however, that if X satisfies the first axiom of countability and x is a limit point of E, then there exists a sequence {x n \ of points of E which converge to x. 4. Let E C. X, where X is a topological space. Prove that x € E if and only if some generalized sequence in E con- verges to x. 5. Let /: S — > X be a generalized sequence in a topological space X. x 6 X is called a cluster point of / if for any open set 0 containing x and so £ S, there is an s > s 0 such that/(s) £ 0. Prove that a topological space X is 1 24 Elements of Abstract Harmon c Analysis compact if and only if every generalized sequence in X has a cluster point 6 Justify the contentions in the one-point compactification 7 Show that if X and its one-point compactification are both H&usdorff spaces then X is locally compact 8 Prove that a compact Hausdorff space is normal 9 Prove that every metric space is normal 10 Let X be a locally compact Hausdorff space Show that the smallest countably additive class generated by all closed sets is the same as the a ring (see footnote on p 125) generated by all compact sets provided the latter contains X References 1 Pontrjagin Topological Groups 2 Bourbaki Topologie Generate CHAPTER 8 Right Haar Measures and the Haar Covering Function In this chapter after drawing on some general results from measure theory, the notions of a right Haar measure and a Haar covering function are introduced. These concepts will then be applied to locally compact topological groups and, in particular, it is demonstrated that consideration of Haar covering functions is nonvacuous, in this framework. Notation and Some Measure Theoretic Resultsf The following symbolism will be used in the ensuing discussion. X: Locally compact Hausdorff space. C: All compact subsets of X. S: The smallest <r-ring that contains C. Analytically, this is the intersection of all v-rings that contain C and is the same as the cr-ring generated by C. This collection of sets, S, will be referred to as the Borel sets. We shall assume throughout that X, itself, is a Borel set. t A collection of nonempty subsets of an arbitrary set, T, having the property that differences, and finite unions of members of the collection are also members of this collection is called a ring of sets. If, in addition, T itself is also a member of this class the collection is called an algebra of sets. (Note that in an algebra of sets the operation of taking comple- ments is closed.) If any countable union (as opposed to just finite unions) is also a member of the class, then the class is said to be countably additive or to be a a-algebra if T belongs to the class, or a a-ring if T does not belong to the class. A measure is a mapping from a ring of sets into the nonnegative, ex- tended, real numbers having the property that the image of the null set under this mapping is 0 (this is to rule out the possibility of defining the measure of every set to be infinite) and the measure (image) of any countable, disjoint union that happens to be a member of the ring is the sum of the measures of the individual sets. 125 126 BemwrtJ of Abstract Harmonic Analysis Definition 1. A mapping ft is called a Borel measure on S if M is a measure on S and the measure of any compact set is finite Definition 2. A right Hoar measure is a Borel measure on a locally compact topological group, G, such that (1) the measure of any nonempty open set is positive, and (2) - „ (E), E 6 S, g € G A left Haar Measure is a Borel measure on a locally com- pact topological group, G, satisfying (1) above and such that tt(gE) *= ft(E) Remark I We note immediately that any statement we can make about right Haar measures implies the existence of an analogous statement for left Haar measures and, for this reason, we will restrict our discussion main!y v to right Haar measures. Remark 2 Property (2) of Definition 2 is equivalent to saying that a right Haar measure is invariant with respect to a nght translation As an illustration of this, consider the additive, locally compact, topological group offered by the real numbers, with respect to Lebesgue measure ft is clear that in this framework the Lebesgue measure of any non- empty open set is positive and, further, that the I ebesgue measure of any set on the real line is invariant with respect to a right (or left) translation Hence Lebesgue measure is a right Haar measure here Remark 3 Property (1) of Definition 2 is equivalent to specifying that ft is not identically zero Proof Suppose there was some open set, 0^0, such that *i(0) = 0 Let and consider Og~ x It is clear that Of 1 is an open set containing e Let V — Og~‘ By property (2) ji(0<r l ) = 0 also 8. Right Haar Measures and Haar Covering Function 127 Now let E be a compact set and let h £ E. Then, since e £ V, h € Vh and we can write E C U Vh. heE Since E is compact we can write EC U Vk { «-i and n(E) < n < 'EniVht) = nju(F) = 0 1 or n(E) = 0. Therefore /x = 0 on all of C and it follows from general measure theoretic considerations that m vanishes on the er- ring generated by C which is S. Remark 4. If a right Haar measure is defined, then a left Haar measure is (implicitly) defined too. Proof. Suppose n is a right Haar measure. Define v{E) = n(E-'), E 6 S. (Note that E £ S=$ E~* € S.) Now v{gE) = niE-'r 1 ) = *(E~ l ) = v{E). Hence v is a left Haar measure, it being clear that property (1) is satisfied. Before proceeding the following definitions and theorem will be necessary. Definition 3. Let X and Y be arbitrary sets and let & and Sy be cr-algebras in X and Y, respectively. A mapping T: X-+Y 128 El emeriti of Abstract Harmonic Analysis is said to be a measurable transformation il, for any set F £ Sr, T~ l (F ) € Sx Further, if n is a measure on Sx then we will denote ti(T~ l (F)) by nT-'(F) This is a measure on Sr, called the induced measure (induced by T) Definition 4. Under the same assumptions about X and Sjr a mapping f *->/?u(±«| is said to be a measurable function f if f~ l (a, «) 6 Sr for any a € R U {±<« ) Using the same notation as above we can now state the following change of variable formula Theorem Let X — * V-*iRuJ±«} where T is a measurable transformation / is a measurable function, ft is a measure on Sx, and nT'~ > is the induced measure on Sr Then, if F € Sr, /(T(x»d M (*) - (fiv) dft(T-Hy)) in J r or fTd M *= / fd M T -» j t <F) J r With this theorem m mind consider G^G^RV{± «} T g —* gh where x is a Borel measurable function and T~ l is given by T~ l g -* gh- 1 t Denoting the smallest * algebra containing all eloecd {open) sets of the real line as the Borel tele of the real line we could restate Definition 4 a-9 follows f tea meaeurable function if the inverse image of any Bore aet in the extended real numbers is in Sj 8. Right Haar Measures and Haar Covering Function 129 the o - algebra in G is S, and g. is a right Haar measure. We now have / xW dn (g) = j x{g ) dnighr J T (G)-0 J n but the last term on the right reduces to / x(g) dfi{g) J G by the right invariance of fi. Thus the right invariant Haar measure has led to a right invariant integral. We will now show the converse is also true ; namely, that a right invariant integral leads to a right invariant measure. Denote by ks(g) the characteristic function of E 6 S, and suppose [ heigh) dfi(g) = f k E (g ) dix(g) = J n J n But Therefore gh £ E =» g € Eh~K p(Ehr') = g(E). The Haar Covering Function Throughout the rest of the discussion the following nota- tion will be used. G\ Locally compact topological group. C 0 (G ) : The class of real-valued, continuous functions over G with compact support. CJ(6): The class of all nonnegative continuous functions over G with compact support. ip: Whenever this symbol appears, even if with a sub- script, it will denote a function from Cf{G) that is not identically zero. 1 30 Elements of Abstract Harmonic Analysis With these notations m mind we can now proceed to the Hoar covering function Consider/ 6 C£(G) and consider all possible finite collec- tions of elements 0u 02, ■•*,{?. e G and nonnegative constants C$, Ci, C, satisfying f(0) ^ Ec,p(gg t ) for all g € G (a) Assuming some collection exists that satisfies (a) we can now define the Haar covering function for f wth respect to as (/?)■= inf £>., i the inf being taken over all sets of nonnegative constants associated with some collection of elements that satisfy (a) ft wifi now be shown that such elements and associated con- stants do exist In any case, since v is not identically zero, there must be some clement h € G such that v’(h) >0 Since this is so there must exist a positive number, «, such that <p(h) > t > 0 Let V — ji? | <p{g) > t) =» ») We can now note two facts about V (1) Since it is the continuous inverse image of an open set it must be open, and (2) U 9 * 0 because h E U Since G is locally compact there must be some neighborhood of A, IT, such that If’ is compact Consider the neighborhood of h defined by V = V n If Now, since f' C If > it follows 8. Right Haar Measures and Haar Covering Function 131 that V is compact. But VC uc { g\ v (g ) > ej SO VC {g ! <p(g) > e}. Hence inf <p(g) > e > 0. oeV Since V C V we also have m 9 = inf <p(g) > inf <p(g) > 0. gtV 0CV Now, since there must exist some compact set, E, outside of which / vanishes and a continuous function must actually attain its extrema on a compact set, there must exist a posi- tive number, M f , such that fig) < Mf, all g e G. For any g € E we can write g € Vh-'g where h is an arbitrary element of V. Letting g range through E, it is clear that the sets { Vlr'g] cover E. Since E is com- pact there must exist gt <E G such that n EC U VgJ 1 . 1=1 We now make the following : Contention. For every g € G " Mr fig) < JL — viggi)- O) «-l m r To prove this we must consider two possibilities: (1) If g 6 CE, then f(g) = 0 and (1) is satisfied. 132 Element* of Abstract Harmonic Analysis (2) If g € B, since the { l g7' I coscr B, there must exist an t such that g € T'g7 ! or gg, € V which implies vigg.) t Adding positive terms to the left hand side, 5Zv>(W.) ^ n * which implies £ WOTO 5, J 1-1 w * Therefore M,£tlnA > m, >/((,) .-I which completes the pnJof Remark 5 This proof immediately implies (/ *) < fiM/ m. In the following theorem we list and prove some rather immediate consequences of the definition of the Haar co\ cr- ing function Theorem 1. If / £ CJ((?) then the following is true (1) Define f k (g) ** f(gk) Then (/* <i) - (/ <f>) (2) If a > 0 then (af #>) = «(/ (3) </, +/, ?) < (/, *) + (/, (4) (/ ?j) < (/ Proof (1) Let g t and c, be such that (a) is satisfied or /W < 2>Wsf.) <«) ( 8. Right Haar Measures and Haar Covering Function 133 and consider Mg) = f(gh) < Xav (<?%)• Thus the elements c it hgi satisfy (a) for f h and we have |c» l/(g) < Xc<¥>(09i), for all gj C di\Mg) < ^Ldi<p(gg<), for all g . (2) We wish to show this also goes the other way around and for this purpose consider jd,}, jg,-J such that Mg) < Udiviggi) => f(gh) < J^di<p(ggi). i i Letting y = gh we have f(y) < Jldi<p(yfr 1 g<) for all y £ G i to reverse the inclusion in (2). Therefore (fh - <p) = ( /'• v ) • (2) If f(g) < J2ci<p(ggi), then af(g) < ^acivigg.) , a > 0 and conversely. Hence ?34 Elements of Abstract Harmonic Analysis (3) For any t > 0 there must exist c„ g, such that for /lCfl) < J2c,<p(gg,) and £c, C (ft v) + t by the definition of in/ Similarly there must exist d,, h, (i =- 1, 2, k ) such that t i /i(f>) < and £d, < (/, v ) + * We non have (/i +/»)($> «/.(?) +/.(g) < I>.Kgg.) + Call di = c*+i Aj = g.+j dk =* c*+t A* = g«+* Hence (/. +/. v») 5! = X>. + XX < (/. o) + (/, v) +2« But, since c was arbitrary this implies ( fi + /* «0 < ( h <f) + ( /. v?) (4) There exist c„ g, such that /(g) < XX^igg.) and 2X <(/**)+« ( 3 ) and dk, g* such that vi(g) < XXv’i(ggi) and XX < (<n «*.)+* ‘ * (4) 8. Right Hoar Measures and Haar Covering Function 135 for any e > 0. Rewriting (4) we obtain vi (gg<) < k Substituting this in (3) yields f(g) < H'''iY J dk‘Pz(gg>gk ) . i k Hence (/:«*) < Halid, t k and (f-v 2 ) < 1 ) 4 - e) ( ( vi ■ Vi) + «) and, since e was arbitrary, (f- <pt) ^ ( f- <Pi) ( vd • Theorem 2. If / G C%{G) and / ^ 0, then (/: <p) > 0. Proof. Consider c; and </; such that fig) '< H c ‘<p(<J0i), for all g £ G. i Certainly we also have fig) < sup <p(g)H c <> for any g, qcG i which implies sup/(ff) < sup <p{g) H c ‘- gcG oeG i Therefore sup / < sup (p ~ Thus for any associated sum, H< c <> SU P // su P V a l° wer bound. Therefore 0 < < (/; p). QED sup <p 1 36 Elements of Abstract Harmonic Anclyiii Consider now some function/# £ C£(G) such that/# & 0 We define l.U) ■ and note the following properties which all stem directly from Theorems 1 and 2 Remark 6 If / 5*? 0, then /,( /) > 0 (A ») = (/ y) (/. v) = (A d If a > 0, then 7,{c/) = a/,( /) Remark 7 /*(/») *= Remark - M/) Remark 9 l/(/# /)</,(/)<(/ /.) To prove this we note that part (4) of Theorem 1 implies (/ v) (A #>) < (/ A) and also that (/ ») > _L_ (/• e) ~ (A /) QED Let U be some neighborhood We will denote those func- tions in CJ(G) that vanish on C U by Fv Fv is not empty To see this apply Theorem 8 of Chapter 7 with E = |p} where V is a neighborhood of g Theorem 3. Let t > 0 be given and Jet/i /j •*•,/« 6 C} (G) with the property that £•-» /,(p) < 1 for all g € G Then there exists 17, a neighborhood of identity, such that for all <p € Fv and any / 6 C# (G) tu/A) </.(/)(!+«) Proof By Theorem 1 of Chapter 7, there exist neighbor- hoods of e, U„ such that for i = 1, 2, • • n i/.w - /.<«') I < «/" 8. Right Haor Measures and Haar Covering Function 137 for g'g~ l € £/,-• Thus for V = we have I Mg) - Mg') I < - , g'g~ l € V n for every i. By the existence statement of this chapter there exist cu and g k such that Kg) < Z ]ck<p(ggk) (5) k and Z c * ^ (/ : v) + v, v > 0. k Since <p 6 Fv then certainly <p(ggk) — 0 if gg k $ U. Hence in (5) nothing is lost by disregarding all those k for which gg t $ U. Denoting by z* k the summation in which only those k such that gg k 6 U we can rewrite (5) as Kg) < Z *Ckv>(ggk). k We now have Mg) Kg) < H*Ck<p(gg k )fi(g ) for gg k 6 u and any i. ( 6 ) However, gg k £ U is equivalent to gig?)- 1 € u which implies \Mg) - Mis 1 ) I < Using this we can replace (6) by Mg)f(g) < T,*ck(fi(g T 1 ) + 13B Elements of Abstract Harmonic Analysis Since adding a few zeros certainly cannot hurt w e can w nte /.(»>/(?) ^ for e\ ery g 6 0 Therefore (/./ «0 < £&(«£') + ‘A which implies £(/./ rt < £ £a(/.(«r‘) + 0 But, since for any g £,/<(g) < live can say £(/./ *) < &(1 + «) In addition however, we had £c* <(/«»)+•', hence dividing through by ( /« <p) (ft ^ 0) gives T (fJ v>) < . (/. *>) ~ (/ 1?) + V (/» *) (l+«) w here v is arbitrary Letting v go to zero yields £l,IM </.(/)(! +«) which is the desired result It is exceedingly important for the later development to note that one cannot Jet t become arbitrarily small while stilt retaining the same v>, * determined If which, m turn, affects ip It is granted that for any t there is some U and, consequent!} , •some <p that will work but the U and <p that work for e t may well not work for an t 3 < «i 8. Right Haar Measures and Haar Covering Function 139 Although the next two theorems we wish to present are rather top-heavy with hypotheses we beg the reader’s in- dulgence for these theorems will play a vital role later on. Theorem 4. The Conditions. 1. G locally compact topological group. 2. h,h, •••,/« € C 0 + (C). 3. v, X > 0. The Conclusions. There exists some neighborhood, U, of identity such that for all <f> € Fv and all X, £ [0, X]. In words: I v of the linear combination is less than or equal to the linear combination of the I v which, in turn, is less than or equal to I v of the linear combination plus some arbitrary positive number. Proof. The left-hand inequality follows in a straightfor- ward manner from part (3) of Theorem 1 and Remark 8. By the second condition, for each /,-, there exists a compact set, Ei, such that /< vanishes on CEi. Clearly then all of the /,• vanish on We further note that E = U"_i2?i is compact. Let h be a function in CJ (G) which has a positive minimum on E. Before proceeding we will prove that such an h exists. Let Y be an open set containing E. Since G is a locally compact Hausdorff space there must (see Halmos 0?]) exist open and compact sets, Uo and Co, such that E C U 0 C Co C V. 140 Elements of Abstract Harmonic Analysis Also, in a locally compact Hausdorff space, in a situation like this (see Theorem 8 of Chapter 7) there exists a con- tinuous function, h, such that h(E ) = 1 and h(CU 6 ) = 0 •which is a suitable function Picking up where we left off, let / - £»./. + <h where <> Oand Kf,/f on E 0 on CE where it is noted that, by the choice of h,fc an never be zero on E We will row prove the following Contention. g t f — X,/, everywhere 1 This is clearly true on E 2 Since both g x and /, are identically zero on CE the contention is proved Since 32xi. X«/. + th we can apply Theorem 3 to assert the existence of a neigh- borhood of identity, U, such that for all tp € Fv 1 - 32i.0-.f-) - £/,(/».) </.(/)(!+') i i i + <!■)(■ +«) 8. Right Haar Measures and Haar Covering Function 141 By the first part of this theorem we also have £l*(X,/,) < (l^IX/i) + el v my 1 + e) and we claim that e can be chosen small enough so as to make + <?l v {h) < V which will then yield IX(x,<7,) < + *■ Some care must be taken in this argument though, because (p depends on e, i.e. for our argument to be valid we must show that I v (h ) and I v remain bounded as e goes to zero. We first note that I v (h) < (h:fo) < » for any <p, by Remark 9 following Theorem 2. Finally, we can write < 2>.(/.:/o) < TMU-fo) < «» ' i / i i t which shows that /„( 2>X./.) is bounded for any <p and completes the proof of the theorem. Theorem 5. The Conditions. 1. G a locally compact topological group and / 6 C£(G) (/ ^ 0 ). 2. U a neighborhood of e such that |/(fir) — f(§) I < e (e > 0) for g 6 Ug. 1 42 Elements of Abstract Harmonic Anatys s The Condos ons For any 8 > « and any $ € Fv ^ 0) there exist Si a ff»6G and positive numbers Cj o c, such that - £ciKW<) j < 5 for all g € G In words this says that in some sense we can approxi mate f(g) Proof Vi e claim that for all g g ( /(a) - <)Hw ') <f(a)Uaa *) < (/(?) ') (1) To prove it we consider two cases 1 gg 1 6 U In this case the validity of the above in equality is obtained directty from the hypothesis 2 gg 1 £ U Since gg 1 $ U => $(gg ') ■* 0 the inequality is again satisfied hence we have it for all g g \\ e now choose a » > 0 such that (/ '('*)•’ < 8 — t (2) where \fr*(g) — iHp -1 ) and assert the existence of a sjm metnc neighborhood of the identity 1 C G such that for g~‘g € T I Ms) - Ms) I < - P> and such that 1C"" here 11 is compact The last statement is valid for the follow mg reasons Since it 6 CJ(C) it must be uniformly continuous (Theorem 1 Chapter 7) i e for any v > 0 there exists a neighborhood of identity 1 ( such that I Ha) - Ha) I < *■ for <r x g £ J ? ( 4) Take 1 1 f) U = I j Since G is locally compact there is some 8. Right Hoar Measures and Haar Covering Function 143 neighborhood of e, F 2 , such that F 2 is compact. Take w = Fi n f 2 c f 2 and note IF C F 2 and therefore IF is a compact neighbor- hood of e. But any neighborhood of e contains a symmetric neighborhood of e (see observation 10, Chapter 6) ; therefore there exists a symmetric neighborhood of e, F, such that (4) is satisfied for g~ x g £ F and F C IF. Since / £ Co'(G) we can say there is some compact set E such that F = \x\f{x) > 0} C E. Since e £ F, then certainly ECU Vg. CtE Since E is compact there must exist g { , i = 1, • • • , n, such that E C U V 9i . i=i By the corollary to the last theorem in Chapter 7 we are assured of the existence of functions, hi 6 Cq(G), such that n ^2hi = 1 on E t=i and hi = 0, on C ( Fg.) . (To prove this consider the one- point compactification of X, X*, which is a normal space in which E is closed because E is compact in X. Now apply the corollary to the last theorem in Chapter 7; note hi vanishes on C(Fg t ) and therefore on C(IFg,), but TFg, is compact so hi £ C„ (G) ) . It is now claimed that hi(g)f(§) (tigr 1 ) - v) < hi{g)S{g)Hggl l ) <hi(g)f(g) OKsir 1 ) + *0 for all g, g £ G. (5) Element of Abifr-ort Harmonic Anolym UA Suppose g £ CE Since / vanishes on CE it is clear that (5) holds in this case Now suppose g $ CE or g € E If g e E, then for some given t, either g € Vg, or g $ Vg, In the latter case then, k,(g) — 0, m the former case, g € Vg.=> ggT 1 € V => gg-'QQT 1 € V =* I iKefl" 1 ) - Mm 7 1 ) | < v by (3) which proves (S) Since (5) is true for each » we have also ZX($)/(y) (MssF') - *) < ZXfpXftyWw; 1 ) ^ £M$)/(y) (iKw -1 ) + »>) Noting that <p vanishes outside U and \f(g) — f(g) ( < « for gg~ x € V, the above statement implies &(«[</<»> - >H(lC) - -/«)] < 2>.(j)/(5)*(ra7') < 2>.«)t(/(p) + «>*(«-> + rf(S)3 (0) lf/(p) ~ < > 0, then it is claimed that (0) yields ((/(g) ~ *)$*(gr l ) - >/($)) < ( f(g ) + + »/{?) ^ Only the left half of (7) will be proved, the right half following in a similar manner Once again the problem is broken up into two parts 1 g € Vg By condition 2 of the hypothesis and our assumption that 8. Righf Haar Measures and Haar Covering Function 145 J(g) — t > 0 we have m > Kg) - 6 > 0=* g € E=> jtfiiffi = 1. »“1 Substituting this into (6) yields the left-hand side of (7). 2. g <£ Ug=* <p{gg~ l ) = 0, which immediately yields the left side of (7) . It is now claimed that for every g we have (Kg) - UM*) - vi r U) < ij(jjl>*(g^- t )Mg)KS)y j < (Kg) + «)/,(**> +vi,(f)\ Here too only the left side of inequality (8a) will be proved, the right side, (8b), following in an analogous manner. Proof of (8a). If /(g) — « > 0. (7) implies U(g) - 0 r(gg~ l ) < »f(g) + XXtfW’farOT)- (9) (8a) (8b) But (see exercise 4 of this chapter) Mg) <M§) for all g implies UK) < UK). We also know that, in general, Uh)=Uf), h feed > and Uh+K) < UK) + UK- Letting g be held fixed, these results when applied to (9) yield (Kg) - «)!„(**) < »Uf) + i^Jjt*(g<rWS)KS)) Elements of Abstract Hannon c Anatyi 1 46 which was desired Noting that (8a) is clearly valid if fid) t < Q, the result (8a) is proved Nov, , since by part (4) of Theorem i, and ur> w> (/ y) (i* v) < u *') u *•>>■ < s - there must exist a {1 < 8 such that u *•> - Dividing (8a) and (8b) by /.(^*) and using this result gives m -« = /<») -<-«-<> s/(?) < r (r ¥M2MbM \ - '\r ) <As) + < + /.(/) r.t*-) < As) +<+«-.- As) + B CIO) We now wish to apply Theorem 4 of this chapter and to simplify matters will list the correspondence of symbols used here with those mentioned in Theorc'i 4 Theorem 4 Here /. 6 CtiG) h,f \ sup i ^*(g<r») /.<*•) s~ e X, 8. Right Haar Measures and Haar Covering Function 147 The fact that “X” < “X” here follows from Remark 9. Noting that Theorem 4 is applicable, there exists some neighborhood of identity, V, such that for all <p 6 Fv, IM*) < E t lfr*(g.'g~ 1 ) /,(**) hVuf) Let + S - 0 IM*) for every g. (H) IM*) Ci in (11). Then we can replace (10) by m - a < TfifTig#- 1 ) < f(g) + 0 + * ~ P = fit) + 5 X or /(fit) - 8 < Ecif(ggri) < f( g ) + s. QED Summary of Theorems in Chapter 8 1. If / 6 Cf(G), then the following is true: (1) Define f h {g) = f(gh). Then (/*: to) = (/: to). (2) If a > 0, then ( af : <p) — a(f : <p). (3) (/i + /s'- to) < (/r- to) + (/* : to). ( 4 ) < ( f'- <Pi) (<Pi'- <Pz) • 2. If / 6 Cf(6) and/ ^ 0, then (/:<?) > 0. 3. Let 6 > 0 be given and let/i, />, •■*,/» € Co" (G) with n the property that XUi/i(?) 2= 1 for all g € (?• 148 Elements of Abstract Harmonic Analyst Then there exists, U, a neighborhood of identity, such that, for all <p 6 Fv, and any / € Cf(G), £/,(//<> < /,(/)(i + <) 4 The Conditions. 1 G a locally compact topological group 2 /«/,•••,/. 6 Cf«?) 3 »>, X > 0 The Conclusions There exists some neighborhood, U, of identity such that r,(l>./.) < < r.( + >• for all <fi € Fv and all € £0, ^!1 5 The Conditions. ] G a locally compact topological group and / 6 Cf (G) (/ * °) 2 C/, a neighborhood of c such that j /(g) — /( g) I < ‘ (« > 0) for ff 6 Ug The Conclusion For any 3 > t and any it £ Fy(\t 0) there exist tfi. gt g* £ G and positive numbers C| Cl, • , c« such that \J{g) - T!.ceHgg,) 1 < 6 for all Q € 0 8. Right Hoar Measures ond Haar Covering Function 149 Remarks 5-9 Inclusive 5. (/:*) < nMj m f 6. If / ^ 0, then I v ( f) > 0. 7. /„(/*) (/»:*>) = (/: y) (M<p) (fo'-v) 8. If a > 0, then I?(af) = al f (f). Exercises 1. Let js be a Haar measure in G. Show that G is discrete if and only if p({gj) ** 0 for some g £ G. 2. If ft is a Haar measure in G and a is any positive real number, show that an is also a Haar measure. 3. Let n be a Haar measure in G and let E\ and E? be two compact subsets of G such that n{E{) — n(E 2 ) = 0. Does this imply that n(EiE 2 ) = 0? 4. Consider /i,/ 2 , <p £ Cq (G ) , and <p ^ 0; prove that/i(g) < Mg) for all g € G implies that I v ( fi) < I p ( / 2 ) . References 1. Naimark, N armed Rings. Especially pp. 360-366. 2. Loomis, An Introduction to Abstract Harmonic Analysis, pp. 113-116. 3. Halmos, Measure Theory, p. 218, Theorem D. CHAPTER 9 The Existence of a Right Invoriant Haar Integra! over any Locally Compact Topological Group Using mainly the results of Chapter 8 it will be shown here that over any locally compact topological group, G, a right invariant Haar integral exists Our first theorem will be devoted solely to proving this statement The integral will be shown to exist only over the class of continuous func- tions with compact support Co(G), however, and this alone is not quite to our satisfaction What we want is a larger class of functions to work over, or, equivalently, a larger class of sets to which the right invariant Haar measure defined by the above integral may be applied Hence after having proven the existence of a right invariant Haar inte- gral over Co(G) we wish to extend it to a larger class f Two avenues of attack are available to us as possibilities for providing the desired extension (1) We might concentrate our efforts on only the integral and not concern ourselves directly with the underlying measure extension that we are brmging about at the same time In particular it is pointed out that all the ingredients necessary for the Darnell extension approach are there and, hence this possibility is at our disposal but is not elaborated upon further here In Chapter 10 more on the 1 integral approach will be said (2) Alternatively we might proceed in a measure theoretic fashion and concentrate our efforts primarily on the exten- t The desire to do this is more easily understandable wlicn one com pares this to the severe limitation of the Rtornan integral of being re- st noted to functions that are continuous almost everywhere with re- spect to Lebe«gue measure and ita extension to the I^beegue integral 150 9. Right Invariant Haar Integral 151 sion of the measure, and in this chapter we shall pursue this possibility. The integral defined in the existence theorem will be used as a springboard for obtaining a measure. An appendix listing some pertinent results from measure theory is included at the end of this chapter for the reader’s convenience. We wish now to proceed to our main result of proving the existence of a right invariant Haar integral on the linear vector latticef of all Borel measurable functions on a locally compact topological group, G. First we will prove the existence of a right invariant Haar integral over the linear vector lattice of all continuous functions with compact support over G. Theorem. Let G be a locally compact topological group. Then there exists a right invariant Haar integral over C«(G). Proof. We introduce the notation {e>}: All those elements in C${G\ which do not vanish identically and S: ip (E [<p] which do not vanish in some neighborhood of the identity, e. It is now claimed that S is partially ordered as follows: Let <pi, <p 2 , 6 S. Then define if 1 Pl > <P2 ! g\Mg ) = 0} d \g\M = o). It is also claimed that <S is a directed set for given any <Pi, <p2 € S, then, since neither vanish in some neighborhood of e, <p\<p2 € S) and Wf! A <pi, <pi<P2 2! e>2' t See p. 159. 1 52 Bementt of Abstract Harmonic Analysis Let / € C*f (G) and consider the mapping S~* R v -/,</) Thus {/„(/)} is a generalized sequence of real numbers and we will show that it is a actually a generalized Cauchy sequence It is claimed that (with Fv as on p 136) {if, for t > 0, there exists some neighborhood of identity, U, such that 1 /,.(/) - /«(/) 1 < « for all n € Ftr) (•) then {/»(/) j is a generalized Cauchy sequence which implies lim /,( /) exists First it will be shown that condition (•) does actually imply that {/,(/)) is a generalized Cauchy sequence and then ne will show that such neighborhoods of identity exist To proie the former contention consider some neighborhood of the identity, U Since we are in a locally compact space we can assert the existence of another neighborhood of e, V, such that the closure of V is compact,} and V C U Further there must exist open and compact sets l/» and C 0 , such that V C Vo C C U (see bottom of p 139) and some con- tinuous function <p, that assumes the value 1 on f and 0 on CUt> Assuming (•) to hold and U to be the U mentioned in (•) then the inequality mentioned in (•) will hold for all <pi, & in Fv Then, certainly, <p € S and I /,*(/) -/„(/)!<« for tp\, <pt > <p where <pi n € S Thus assuming (•) docs imply that I /*(/)) is a generalized Cauchy sequence We will now prove that ( •) holds If /, fa € Ct(G),f 0 & 0, then there exists some neighbor- hood of e, V, with compact closure, such that I /(p) — /($) | < <i and \Mo) -/»($) I < ** t This follows from the local compactness of O and exercise A of Chapter 6 9. Right Invariant Haar Integral 153 for all g 6 Vg where a is an arbitrary positive number. Since, in the subsequent discussion, we will be interested in “small” values of ex, we will assume now that ex is less than one. Now let t/ € Fv be not identically zero. By Theorem 5 of Chapter 8 there exist c, and g, such that w(g) = | /(g) - 2 Zc<t(ggi) | < 2ex. i where i ranges from 1 to n. By Remarks 5 and 9 of Chapter 8 we can say and I v ( w ) < («>:/o) ( w: fo) „ max to, < k < m f . where k is some integer and m !s is inf / 0 (over some open set) which is positive./ We would now like to show that k does not depend on ex (for all «i < 1 that is) . Once this is estab- lished we will have shown that I v (w) is bounded as e t goes to zero. Since k depends only on the compact set outside of which w vanishes (see p. 131), if we can exhibit some com- pact set outside of which all w{g) vanish, regardless of ex, then we will have demonstrated the independence of ei and k. To this end consider F = \x \ /(*) > 0}. Since / £ CJ (G) it is clear that there is some compact set, E, such that F C E, and, further, the g, mentioned before must be such that EC U Vgr> by the proof of Theorem 5 of Chapter 8. Hence for each i, Fgv 1 n E 0 t See the original discussion concerning the nonvacuousness of Haar covering functions, pp. lpO-132. 154 Elements of Abstract Harmonic Analyt s which implies lg 7 * n E j* 0 We can now assert the existence of a i 6 V such that *87* * x € E=* 177 s - vr'x 6 f-'E or 07 1 e V-'E It is now contended that all to(p), for all «i < 1, vanish out- side the compact set Vi V?E U E where V\ is the neighborhood corresponding to «i » I, and we will prove this by showing that each of the terms, /(<?) and S» C -V'({?P«) is zero outside this set Clearly if g <£ ViV^'E U E then gi E which implies /(ff) = 0 We would now like to show that a sufficient condition for each of the f(gg t ) to vanish is that 9 $ Yjtft 1 To this end suppose 9 § Vi07‘ Vtf? =* f?0. t l r i Since jfr 6 Ft-, then y p(gg t ) = 0 if p $ Vi97 l In our case we have g f \ t V?E, but, since 07 ‘ € Tr 1 #, then 17 i I iS7‘ Therefore Hggt) - 0 9. Right Invariant Haar Integral 155 which implies w(g ) = 0 outside ViV^E U E, and we have established the required independence of k and e t . Before proceeding to our next step in the proof we wish to note some facts which come from the theorems in Chapter 8: 1. It is clear that there is some compact set, E', outside of which both / and f 0 vanish. 2. In Theorem 5 some functions, hi, were constructed. With regard to this construction (with respect to E') it is clear that the same hi will work equally well for f and f 0 . 3. By Theorem 4, there must exist some neighborhood of e, U, such that for all <p £ Fu h(ll c d{ggi?) < = (Zc,)/ v W < I 9 (l ™ (1) where we have the following correspondence: Theorem 4 Here ei/w/c h(hl) sup * = sup — _ yjuj) Ci 1M*) tiggi) 4. It is clear that an analogous statement to (1) can be made about /o because of the restriction imposed upon V; the same U and the same ^ will work for /o, the only change v X X, Si 156 Elements of Abstract Harmonic Analysis necessary being fo replace the c, by This done we will now perform some manipulations with (1) knowing that, in the end result, we need only replace the c t by d, and / by / a to get the same result for /„ By the definition of u>(g) we have for any <p € Fa, that 1,(1) - — l < < 1.(1) + — t mj, \ , / tn/, Using this and (1) together we have i.d) - < (&)/.<« < un + — i + — m/, m,. for v € F v Let = (2k + l)ti/fn f . We have 1 ,( 1 ) - « < ( ) < 1 .( 1 ) + o (2) But, as previously noted, we now also have 1.(1,) - « < (£ijl.(*) < 1.(1.) + ( 3 ) Using (2) and (3) together now gives o - «> - « < i.u) < g~o + ..) + ■> Adding 1 throughout and assuming <* < i we have X>. „ /,(/) + i zj , + ,s ' f-v < 2 (/,(/) + 1) < 2((/7»> + 1) for any <p € Fa. 9. Right Invariant Haar Integral 157 Hence if ^2 € F v , I !«(/) - /„(/) I < 2^1 + = 2(2 k + 1)— (2) ((/:/,) + 1). Letting ei — » 0 now yields the desired result; namely that {/*(/) } is a generalized Cauchy sequence and, since R is a complete metric space, lim I v (f) —1(f) exists, s This done, we will now show that 1(f) is indeed a right invariant integral over Cf (G) to complete the proof. 1. In general if / £ CJ(G) and / N 0, then '• (/) a zb) > 0 for all <p. This implies that 1(f) >0 if / > 0 and /NO. 2. Since, if h is fixed, I'UW) = I,(f(g )) for all <p we have that I(f(gh)) = I(f(g)) as well. 3. If ci, Oi > 0, and v > 0, there exists some neighborhood of e, U, such that for all <p £ FV I f«»(ci/i + C 2 / 2 ) — cil r (fi) — cfl v (fi) | < r by Theorem 4 of the last chapter. Thus 1 f( c i/i + C 2 / 2 ) — cil(fi) — c*l(h) ) < v where v is arbitrary. Therefore /(Cl/l + C 2 / 2 ) = cj(fi) + (hi (ft)- 158 Element* of Abstract Harmonic Analysis Finally if we take J + (g) = and / Iff) - and then take m 0 -Jiff) 0 if f(g) > o otherwise if f(g) < o otherwise /(/) -/</+)-/</) we see that 1(f) ail] hate the properties mentioned aboie over all of C 9 (G ) This completes the proof Haling constructed the aboie integral oter C»(G) our de«ire now is to extend it and as pointed out earlier manj approaches present themselves First we will bneflj show that all the ingredients nccessarj for the Darnell extension approach (as used by Loomis [1] in Chapter 3 of his book for example) to extend the integral from just C 9 (G) to the Batre classes are present The Darnell Exlenuon Approach Noting that Co(G) is a real vector space we are justified m calling the I ( /) defined above a linear functional Further since 1( f) has the additional propertj that / > 0 =>/(/) > 0 we will call I a posihie linear functional Consider now a sequence of functions fi /« of C«(C) that contergcs monotomcaH> decreasing!} to zero f Since /.</.) < </. /.> < — ^ m, t It is clear by D n « theorem that s nee the /. are coot nuotis and have compact support that they must converge un forroly 9. Right Invariant Haar Integral 159 where k is some integer, which can be chosen independently of n, it is clear that I v ( /„) converges monotonically to zero which implies that /(/„) converges monotonically to zero too. Hence monot monot fn >0 =»/(/„) * 0 and we shall refer to this as property (M). We define now /A 0 = inf(/, g) fVg = sup (/, g) where/, g £ C 0 (G). It is clear that the functions / A g and / V g are each members of Co(G). Denoting the above operations as the lattice operations we can paraphrase our last result by saying the Co(G) is closed under the lattice operations or that C a (G) is a linear vector lattice. We can now summarize these observations about C 0 (G) : 1. C 0 (<?) is a linear vector lattice, and 2. 7 is a positive linear functional on C 0 (G) with property (M). These facts will now allow us to define the Daniell exten- sion of 7, I e , to the Baire classes. Further since, denoting the identity function of Ca(G) by simply 1, for any/ £ C 0 (G) / A 1 € C 0 ((?) we are assured of the representation Uf) = [fdp J o where the integral is taken in the customary sense, f This concludes our discussion of the Daniell extension ap- proach and we will now turn toward a more measure oriented procedure. t Loomis £l], p. 35. 160 Element* of Abifraef Harmonic Analysis A Measure Theoretic Approach Before beginning please note that most of the pertinent terms used here are defined in the appendix First the genera! procedure will be outlined and then it mil be applied to the special case under consideration here, but before that wc will need the following Representation Theorem. f If J(J) is a positive linear func- tional on Ce(G), then there exists a unique Borel measure, n, such that JU) = [fd* J o Although this theorem will not be proven here, we will indi- cate how the (nght invariant) Borel measure is constructed from the / ( /) defined before W e first define for any E £ 6, the class of all compact seta in G, \(E) = inf /(/) /«a* Bs - \S 6 CJ(G),| j(g) > k M {g) for all g 6 (?) and ke is the characteristic function of E We state without proof that X is a regular content on C Any content however, induces an inner content as follows Denote the class of open sets of G by 0 and take, for 06 0 X.(0) * sup X(F) rco r«f Loosely, the "biggest*' compact set in 0 is desired here X« is now our induced inner content on 0 This done we can now define an outer measure a*, on G Suppose E C.G Take m *(E) ~ inf X.(0) XCOttf t For this result and for the rest of the chapter a good source is IfaJmoe C3J 9. Right Invariant Haar Integral 161 Loosely, here, we wish to “approximate” E by an open set, Next we define the n*- measurable sets to be those sets, E, such that n*(A) = (A n E) + n*(A n CE) for any set A C G.f In other words the /immeasurable sets split all other sets additively. It can be shown that the n*~ measurable sets contain all the Borel sets, S, and it can further be demonstrated that when one restricts p* to S, calling the restriction fi, that a regular Borel measure is obtained. Diagrammatically the process is as follows: 1(f) over C 0 (G) \ X over C \ X. over 0 \ n * over all subsets \ n over S where the arrows are to be interpreted as “gives rise to.” Further the measure p so defined is right invariant by the following : Theorem. Let X be a locally compact Hausdorff space and let T be a homeomorphism where T: X->X. Then, if X is a content, X(F) = \(T(F)), F compact, is also a content and the corresponding measures have the property that fi(E) = n(T(E)) for all E £ S t Please note again that we are assuming the entire space, G, to be a Borel set. Otherwise. 1 , we would have to consider open Borel sets, and consider p* on the hereditary cr-ring of all <r-bounded sets. 1 62 Element! of Abstract Harm ante Analysts where n is the measure determined X and p is the measure determined by X In our case we note that since I(/») = /(/), where M§) *= Sigh) then \(Eh) = \{E) for all E 6 C by the definition of A Consider now the homeomorphism T G—*G Q-*gh and consider X(fT) =■ \(r(£)) = A {Eh) = A (E) By the above theorem we can now say that ME) •= u{E) ~ n(Eh) where A —* n and X — * p which establishes the right invariance of n It is to be noted that the fact that A is a regular content guarantees for any compact set F that ME) = A (F) so that v is indeed really an extension of the A we started n Uh In summary \\c have obtained the following 1 A Borel measure that is right invariant or equivalently 2 a measure that is right invariant on all Borel sets 3 Hence the corresponding integral is right mvnriant on all Borel functions Even though the following two facts have been implicitly established by the preceding discussion we shall prox e them here to be able to insure that our exten sion is a right invariant Haar measure (A) If 0 £ 6 then MO) >0 provided 0 ff (B) If E is compact then ME) < « Proof of A Let 0 € 6 0 0 Since 0 9*0 there must be some element g € O Since G is regular (see exercise 4 of Chapter 6 and p 152) there is some neighborhood U of g such that gtVCO where V is compact Then, as we know, there must be a 9. Right Invariant Hoar Integral 163 tinuous function/ such that 0 </< 1 1 where / = 1 on U and / = 0 on CO. jarly we have ko(g)>f(g), all g. nv since / ^ 0 we have 0 < 1(f) = f f dfi < (k 0 dti = „(0) J G J G hich proves A. Proof of B. Let E £ C. Recalling the proof of Theorem 4 f the last chapter, there is some continuous function h € 't(G) such that h = 1 on E. 't is clear that h > k E , the characteristic function of E, But f hdn = 1(h) < <» . •'o Therefore »(E) = f k E dn < (hdti = 1(h) < «. QED ~ G ^ G Appendix to Chapter 9 Definition 1. A real-valued mapping, X, is said to be a regular content on C, the class of all compact sets of any topological spacef if 1 . E e <?=> 0 < \(E ) < 2. E h E 2 e C and E x C E 2 =>\(Ei) < h(E 2 ). t Although we can define a regular content on any topological Space, the most interesting applications arise when the space is a locally com- pact Hausdorff space. 144 Clemen ri of Abstract Harmon c Analysis 3 E x E t € d and E x ft F* = p=> X(£, U E t ) « X(£’ 1 ) + X(F,) 4 FT, F, € £=> X,(F, U £7,) < \(F,) -f X(£7,) 5 (Regulant)) Denote the set of interior points of a set F by F° and let £ € C Then X(F) = mf{X(F) \ E C F° C F € C| Definition 2 The induced tnner confenf induced by X X« is given by X*<0) « sup|X(F) | F C O, F e Cj where O^O Definition 3 let f} be a hereditary c ring The mapping n* fl — » 5 U | -f-°° } is •'aid to be an outer measure if (1) is* > 0 (2) n, ■ H. i (3) (Monotonicity) If £i Ej € tl and E\ C Fj then h*(£.) < M*(&) (4) m*(0) - 0 Definition 4 5a «ubsct of a topological space is said to be o-ixmnded if there exists a sequence of compact sets {F«|, such that BCU£, Exercises 1 Pro\ e that e\ ery Borel measure is a finite i e if a is a Borel measure and if E is an) Borel set, then there exists s sequence {E n f o! Borel sots such that F d lf%}E» and n(E m ) < « 9. Right Invariant Haar Integral 165 2. Let fi and v be two Borel measures such that v{B) = 0 whenever y(E) =0. Prove that, if y is regular, v is also regular. (See footnote on p. 215 for the definition of a regular measure.) 3. Let G be a locally compact topological abelian group. If n is a Haar measure on G, show that y{E~' 1 ) = n(E), for any Borel set E. References 1. Loomis, An Introduction to Abstract Harmonic Analysis. 2. Naimark, N armed Rings. 3. Halmos, Measure Theory. cHAprfR ro The Daniell Extension from a Topological Point of View, Some General Results from Measure Theory, and Group Algebras In this chapter an alternative approach to extending the integral defined in Chapter 9 will be illustrated Essentially it is just the Darnell extension but from a topological point of view Next it will be shown that the integral defined over Co (G) m Chapter 9 is unique to within a constant multiplier and then some examples of right Haar measures will be given We will then sketch some general results from measure theory and state the Fubim theorems used in Chapters 1 and 2 m the genera! situation Final!) we will include some discussion of group algebras Extending the Integral Suppose A is a locall) compact Hausdorff space and denote bj Co (A ) all those complex-valued continuous functions, /, over X with compact support £Wc note that Co(A') is a linear vector lattice such that, if / g 6 C, {X), then/ A 9 € C*(X) where as usual, / A g denotes the function defined as inf( f(x) g{x)) ] In addition, suppose L(f) is a positive linear functional defined on Co(A') fSmee L{ /) is a positive linear functional it immediately follows that (£(/)! < win Definition An extended real-valued function g is said to be lower semieonUnuous if for an) xa £ X and an> t > 0 there is ttfvgkbsrhaod of a ft {* 9 ) s nmJ? tint /©r^D -S' 6 g{*) > oM - € J66 10. The Daniell Extension 167 We will denote by M+ the set of all nonnegative lower semi- continuous functions. It can be shown that for any / £ M+ that / = sup Y, = sup {<7 € Ct(X) \ g <f\. Putting this in a very loose way one might say that it is possible to approximate any lower semicontinuous from below by a continuous function. We now have our first ex- tension: for any / £ M+ we define L*(f) = sup L(g) and it can be shown that L* is almost a positive linear func- tional on M + ] namely, L* is additive, L*(cf ) = cL*(f) if c > 0 , and L* is positive. To extend this further let / be an arbitrary extended nonnegative function and consider Z f ={ge M+\g>f\. (Clearly Z t ^ 0 because g = «> is in Zj.) We can now define L,(/) = inf L*(g). BtZ/ Lt is unfortunately only subadditive, however (over the class of nonnegative functions) ; i.e., if /i, / 2 > 0, then L*(fi -h/2) < + L»(/ 2 ) in general. Actually L* is countably subadditive. Given L* though, we can define an outer measure, n*, as follows: Let A be a set and let k A be its characteristic function. Takef m*(A) = L.(/c a ). We will call A a zero set if ti*(A) = 0 . This done we can now introduce the following equivalence relation among the functions defined over X : /1 ~ /2 if and only if /i = / 2 everywhere except on a zero set. t It is left as an exercise to verify that A is actually an outer measure. 168 Element* of Abstract Harmonic Analysis Consider now the space of al! equn alence clashes of complex- valued functions,/, such that Mi/i) < * and call this class 1/ If w e take as a nonn on L 1 ll/ll = MI/I). it can be shown that L l is complete with respect to this norm Also, by a previous observation, | L(/) j < (1/ 11 for / C.(X) Consider now the closure of C t (X), C#(A),and call this class Li Li is a closed subspace of L l and, b> virtue of this observation, it follows that In is complete (A closed subspace of a complete space is complete ) Denotin g the continuous extension of Lover C#(X) to L.over C«(A) = Li w e w ill call a 6Ct A summable if and onlj if k A £ Li Using this, however we can actually define a measure, n, on the class of summable sets by taking /i(A) = L.(U We will call a 6et A, measurable if the intereclion of A with anj compact set is summable and it can be shown that /( L.U) > j (/* < » Conversely it can al«o be shown that j fd(t < «>=>/€ Li and L,( /) = j f J d m One can also show that all open and closed sets are meas- urable Diagrammatical!.} what ire have done can be summed up as follows (subadditiv e) L.(/) anj nonnegative/ I MHav* Jf + I L(/) over C.(X) 10. The Daniell Extension 169 where the lines should be read upward as “has been extended to.” Using the L»(/) we then got L'(f) over C a (X) C L 1 [via L.(/)] L{f ) over C„(X) where L r ( /) is the desired extension and completes our general outline of this extension procedure. In the special case where X — G is a locally compact topological group and L is right invariant on Ca(X), it can be seen readily that L e is right invariant on Co(X). We can now proceed to discussing the uniqueness of the /(/) [over Co(<?)D defined in the preceding chapter. Uniqueness of the Integral As usual, G is a locally compact topological group and whenever G is used in the remainder of the chapter it will be assumed to have those properties. In the preceding chapter the right-invariant Haar integral, 1(f), was defined over Co(G) and we now wish to suppose that another right- invariant Haar integral, J(f), has also been defined over Cii(G) . If we can show that ./( /) = al(f), where a is con- stant, on C b (G) then, .by any of the extension procedures, we can show that (denoting the extensions of I and / by I and J) J(f) = o.I ( /) on all Borel functions. To this end let /,HOT, v^O and choose d, and g< such that f(g) < for all g. (1) This implies that j(f) < ' 1=1 or that J(f) /J(f) is a lower bound for all 23>d>- Hence J(f)/JW < (}■■*) 170 Elements of Abstract Harmonic Analysis or J{f) < U *)J(+) (*) Consider some function, fi € Ct(G), such that ^ 0 Bj Theorem 5 of Chapter 8 there exists some neighborhood of the identity, T, such that for all ^ € fY t»(g) - l/i(ff) - TfsHsa.) I < 2»i (2) where <i is an arbitrary positive number, $ is assumed to bo not identically zero, and the c, and g, are as m that theorem Suppose now that »i < j and recall that there must exist a compact set, Ei, such that I g\ft(9) > 0 } c Bt Further (See proof of existence theorem in Chapter 0) all id's must vanish outside the compact set tM'r i £i u Ei where V\ corresponds to <i => J We can now assert the exist- ence of a function /* £ Cf(G) independent of <j such that (see proof of Theorem 4 Chapter 8 where such a function is mentioned) /, - l on iVf'EiUF, This and Eq (2) imply A(sr) + 2<./,(»> > O) and also Ma) S 2*1 ft(p) + £t*(w.) « I rom (3) it immediately follows that Jifi) + 2 .,/(/,) > < 3 «) while (4) implies that </»*)- &,(/, *) < (4a) 10. The Danieil Extension 171 Multiplying through in (4a) by J(\f>) and using (4a) in (3a) we have J(h) + 2ftJ(A) > (l - (hU)JW- (5) Dividing (5) through by /(/) and using (*) we obtain J(h) JU) + 2ej JU*) JU) > (1 - 2 £l (/ 2 :/,)) hUd hU) ' (6) Taking the generalized limit over t p first, and then letting ei — > 0, we have J(h) > iuo JU) ~ Kf) ' But, since / and fi are interchangeable we also have JU) > KR J(h) ~ Kh) which implies Kf) JU) for any / £ Cf(G), and, consequently, for all / 6 C 0 (G) . Calling the positive constant 1(h)/ J (h) — a we have /(/)=«/(/) for all feC»(G), or equivalently mi = a MJ where mi and mj are the right-invariant Haar measures deter- mined by I and J, respectively. This completes the proof. 172 Semen tj of Abjtroct Harmonic Analysis Example* of Haar Measure* Example 1 ff, + («e the additiv e group of real numbers) , n Lcbcsgue measure right invariant integral j j{x) dx Example 2 R*, • (le the multiplies tn e group of non- zero real numbers) , n Take, for any Bore! set E C R*, J t x where the integral is taken in the Bebesquc sense To prove n is right-invariant consider X<«) - / 7 dx d(xt) ( dx As nght invariant integral we can now take / /(*) *(«> J R* Example 3 C* Take for any Bore! sct,£,in C*, / dr where v represents two-dimensional Bebc«gue measure ,,, . / '!• r ii.r<fr (Note that the Jacobian of the transformation is just | U ]’ here ) Hence m is nght in\ anant and n e have as a right- jmaxaat miegjsd f mdMd) J c • 1 0. The Daniel! Extension 173 Remark. In each of the above examples the underlying groups have been abelian which implies that if p is right- invariant it will also be left-invariant. Further, in any abelian group we also have ti(E) = y.{E~') (by exercise 3, p. 165). In our next example a nonabelian group is con- sidered. Example 4. As the underlying multiplicative group G consider all matrices ( x y\ 0 < x < oo 0 1 / —00 <?/<«> and take as a topology the following: It is clear that a 1-1 correspondence exists between the points ( x , y ) of the open right half plane and all matrices of the form above or there exists a mapping x y\ i-i (x, y )■ As a topology on the space of matrices above we will take the associated topology from the plane. This done we will take, for every Borel set E in G, and v{E) = u dx dy E J X Now let £ = (a b\ VO 1 ) We note that this implies that the elements of %E are of the form where (x y\ VO 1 J 6 E. 174 ETemenfj of AbiJroef Hormomc Analyst In the plane this corresponds to x — » ai y -» ay + 6 so -<® -/„/*? because the Jacobian of the transformation is just a 1 Hence (i is left invariant Consider £$ now The elements of /£ are of the form whieh corresponds in the plane to x -*ax y -* bx + y which has the Jacobian a Hence wU**- and w e ha\ e established the right im anance of r Consider now CD'-CT) and note that 1 0, The Daniel! Extension 1 75 which corresponds to x — > 1 x Hence ^ i} = fJl dxdy=i,(E) - It is clear now that there exists E such that n(E) < oo but v(J?) = m^' 1 ) = «. Our next theorem gives us an equivalent way of deter- mining whether a topological group is compact or not. Theorem. G is compact if and only if n(G) < a> for any right (or left) invariant Haar measure, p. Proof ( Necessity ). Suppose (7 is compact. This implies that the identity function, 1, has compact support or 1 6 Co(G). We now have m(G) « dp = 1(1) < =0. J a ( Sufficiency ). Suppose (7 is not compact and let F be a compact neighborhood of the identity. Since the finite union of compact sets is itself compact then we must have G s* U Vg< i- 1 for any positive integer p. J 76 Element of Abifroct Harmonic Anatyui Choose elements, gi, g%, now such that ?.{ U Vg t i-i and let U be a symmetric neighborhood of identity such that IP C V We now claim that the sets {£/».} an; disjoint Suppose not, then (assuming that n > m) Ug, fi Ug m / 0 s=> there exist ui «, m U such that «i g, ~ «*?» =» g, = f IPg* But I Pg m C Vq<* Hence we have contradicted the very way in which the g K were chosen Therefore the sets \Ug.\ are disjoint Now wc have O O U Ug, which implies n(<7) S =* Iim » 00 which completes the proof Remark If the above theorem applies then the Haar measure is usually normalized «o that n(G) *= 1 Product Measure* Consider two sets X and Y and corresponding i rings of sets Sx and Sr Let ^ be a measure defined over Sx and v be a measure over Sr What we would like to do now w define a measure over some collection (<y-nng) of sets in A X r To this end consider Sj X Sr the a-nng generated by all A X B where /l € Sx and B £ Sr e would now like to state some results from measure theory but before doing so two new definitions are necessary Let f be a ret m the product space X X I , the set F, “ iy[{x,y) € Ft 1 0. The Daniel! Extension 177 is called an x-sedion of E. Similarly the set E v = (*| (*, y) € E) is called a y-sedion of E. Consider some real-valued function, f(x, y), over X X Y now, and consider the function f x : E X —*R where f x (y) = f(x, y). The function f z is called an x-sedion of f. Similarly the function E v — » R where /„(x) = f(x, y) is called a y-sedion of f. We can now avail ourselves of the following results from measure theory. In the following statements X and Y will be sets, Sx and Sr will be <r-rings of sets in X and Y, respectively, and fi and v will be <r-finitef measures over Sx and Sy, respectively. 1. If E € Sx X Sy, then every E x € Sy and every E v € Sx- We can reword this as follows: if E is measurable in X X y, then every section of E is measurable. 2. If f(x, y) is a measurable function, then every section of / is measurable. 3. If E is a measurable set in X X Y, then the functions p{x) = v{E x ) and q{y) = y(E v ) are measurable functions over X and F, respectively. Furthermore f v{E x ) = f p(E v ) dv(y). •' X Y 4. Let E be a measurable set in X X Y. Then HE) = fv(E x ) dp(x) = f HE,) dv{y) J X J y t In this context a measure over a ring of sets, S, will be called tr-finite if for any set E £ S there exist E n € S, n = 1, 2, . - such that E <= Un-i E„ where (S„) < for all n. 178 Element* of Abitract Harmonic Anolyjlj constitutes a a finite measure on S. r X Sy and if A (■ Sx and B € Sr, then X(A XB) * n(A)v(B) The measure X is called the product mcajurc and is often denoted by X = n X * One also notes in passing that the propertj X(A X B) = *(A)»(0), X € Sx, B € S r determines X uniquely over Sx X Sr Having defined a product measure we can non talk about integrals over measurable subsets of A' X Y In the same notation as the preceding we shall call the integral f h(x y) dX « f hix, y ) d(p X >) J xxr J xxr a double integral Just as in the case of double or tnplc inte- grals in euclidean space we turn our attention next toward titrated integrals To this end consider f(x) *= f h(x y) <My) - / My) <M«/) J r J r assuming of course that the integral of My) exists If Jx f(x) dn also exists wc wntc f fix) dp = f f hix y) drdu = f duf h(x, y) dr, J x J x J r J x J r and call this an iterated integral We can, of course, inter- change the roles of X and F in the above discussion and consider ( dr f hix, y ) d,i J r J x assuming it makes sense \\ ith these v cry natural generaliza- tions of concepts famifiar in euclidean space in mind, we can now state the generalized v eroons of the Fubim theorems TO, The Darnell Extension 179 we made such extensive use of in Chapters 1 and 2.f To simplify matters later on when we wish to refer to these theorems we shall call them simply the Fubini theorems. The theorems overlap in content, and we state them all simply for convenience. Theorem (FI). If h > 0, and measurable on X X Y, then I h dk = I I h dv dfi = I j h dy dv. •'t-W j y J v J v J y Theorem (F2). If h is integrable on X X Y, then almost every section of h is integrable which implies f(x) = [ h(x,y)dv(y ) = f h x (y) dv(y) J r J y exists a.e., and also that g(y)~ = / h(x, y) dy(x) = / hy{x) dy{x) J x J x exists a.e. Also /(a;) and g(y) are integrable, and f fhd\ = f j h dv dy = j j h dy dv •'wv * J x ' Y •'* v •'Y 'XXY 1 Theorem (F3). If h is measurable on X X Y, and one of f | h | dX, f f | h j dv dy, f f \ h\dydv J XXY " v “ v ^ v ^ Y ’ Y ’'X exists and is finite. Then all three exist, are finite, and are equal, and f h d\ — f f h dv dy = j J h dy. dv. ■'xxy •'x-'r J Y J X t See Theorems 3 and 4 (Fubini’s theorem and the Tonelli-Hobson theorem) of the appendix to Chapter 1 on pp. 16-17. 1 80 Element* of AbUract Harmonic Analysis Remark It can be shown that any Borel measure is <r- fimtc (see exercise 1, Chapter 9) and, further, that if n and » are two regular (see footnote, p 215) Borel measures over the Borel seta Sx and Sr, respectively, then u X r is also a regular Borel measure Suppose now that X and Y are locally compact Kausdorff spaces and that ft and r arc regular Borel measures defined over the classes of Borel sets S x and Sr, respectively, where, of course, Sx is a class of subsets of X and similarly for Sy By virtue of the remark, tn any framework of this type, fi X v must be a regular Borel measure With this in mind consider a locally compact topological group, G, and a right Haar measure, n Denote by Li(Q) all equivalence classes of complex-valued Borel measurable functions, /, such that f 0 \f\d>i < « As a norm on U(G) we can take ii/ii, - / m* We can show that L\(fi) is actually a Banach algebra with respect to this norm if we take as multiplication / * g, where (/•?)<*) *= / /(* y~ l )g(y) My), f,g £ h(G) J a In order for the above definition of multiplication to make sense, however, we must show that the integrand is measur- able, and that the integral exists for almost every z In order that Lj{G) be a Banach algebra we must then show that /•g € Bi and l[i < ||/jh jiff lit* and we shall proceed to the demonstration of all these facts now Let 0 be an open set in the complex plane and let E ** /“•(O) Denote f(zy~ l ) by h(x, y) If we can show that k-'fQ) - |<ar, y ) \Jfzy~') Z 0 or zjr' Z f'HO) - E) is a Borel measurable set, then we will have established that h is a measurable function and, since t c(z, y) ** q(v) w 10. The Daniell Extension 181 clearly Borel measurable, we will have that the integrand, is a Borel function on G X G, and thus every section is measurable. It is clear that the set Ei = E X G is a measurable set in G X G. We would like to show that h~ l (0) = E 2 = [(x,y) 6 (? X G | xy r 1 6 E j is a measurable set in G X G. To this end we note that since homeomorphisms preserve compact sets and, hence, Borel sets, that if we can show Ei to be a homeomorphic image of E x we will have estab- lished the measurability of E 2 , because we have already noted that E\ is measurable. Proceeding according to this plan of attack consider the following homeomorphismt: G X G G X G (x, y) ->• (xy, y) Let (x, y ) 6 Ei. By the definition of E%, x 6 E and y 6 G. Now since xyy~ l ~ x € E it follows that (x, y) has its image in E 2 . Conversely suppose (x, y) € E 2 => xy~' € E=> (xy- 1 , y) 6 Ei. But (xy- 1 , y ) -> (xy- x y, y) = (x, y) . Hence for any point, (x, y), in E 2 we found a point in E t that maps into it. Thus we have established the fact that h is a measurable function, and also the measurability of the integrand in question. To show that the integral exists for almost every x, consider f dy(y) f \Kxy- 1 ) 1 1 g(y) | dy(x) J G J G = f \g(y)\dn(y)f {/(xy- 1 ) \dy(x). J a j g t It is easily verified that this is a homeomorphism by using Axioms G3 and G4 for topological groups listed in Chapter 6. 1 82 Elements of Abstract Harmonic Analysis Using the right invariance of » m the last member wc ha\e that the above product is equal to ll/HilUII. By Theorem F3 of this chapter, [ /tar‘)ff(y) d(n X n) j gxg exists and is finite Also by Theorem F2, [f(xy-')g( v) My) J o exists for a c x and is integrabic Therefore / • g 6 1* Now we must show that 11/ *Pl!i 5 11/ Hi 11 9 jit The existence of II/* fill - / I (/•*>(*) I <W x) J a has been established Expanding this wc have 11/ • g Hi = f <M*) I f f(xy~')g(y) My) I J o J a < I dp(z){ |/(iy-')p(y) l<fr(tf) J a J a Interchanging the order of integration in the last term we have that it is equal to I 1 ff(y) I ifci(y) j i/(r tr l ) I <&»(*) J o J tj which, bj the right invariance of u equals ii/ ii> iip ii. < - 1 0. The Darnell Extension 183 which justifies the above interchange in the order of inte- gration and establishes the desired result. Thus L X {G ) is a Banach algebra. Theorem. Let G be commutative then f *g — g* f. Proof. By definition (f*g)(x) = J f(xy~ l )g(y) dy(y). Replacing y by yx in the integral and using the right invari- ance of u we have the above integral equal to ff(y~ l )g(yx) dy(y). Replacing y by y~ l and noting that G being abelian implies that n(E) — n(E~ l ) we have U*g)(x) = J f(y)g{y~ l x) dn(y) = Jg(xir l )f(y) dy(y) “(»*/)(*). QED It can also be shown that the converse of this statement' is true (see exercise 1). An interesting reflection of whether an identity is present in L X {G), in the topological character of the underlying group follows. Theorem. Li(G) has an identity, if and only if G has the discrete topology. Proof. Suppose first that G is discrete. In this case the one-point sets of G are open sets, each of which have the same positive measure, which we may assume to equal 1. 1 84 Element! of Abstract Harmonic Analysts Symbolically, if ne denote the nght Haar measure by n and let g be an element of G we have v{\o)) - 1 This convention then implies //<*) *(*) - Em ■'o icO which means that /is integrable if and only if / *= 0 cieiy- where except on some countable set Xi, x t , • • • and 52 l/(*.) I < « Now consider (/•ff)(*) - fj(xy~ l )g(v) dn(y) - 52/(*jr‘)y(y) ytO and take 1, x ■= e e(x) >= 0, x He where e is, as usual, the identity clement of G Wc now ha\ e (e •g)(x) = 52*fo r‘)p(y) - p(*) and can, in a similar fashion show that ( g • t) (x) — g(x) to establish the sufficiency of the condition Convene Suppose there exists an identity, m, for L%{G) We now contend that this supposition implies that the set of all measures of nonempty open sets in G has a positive greatest lower bound Suppose not, this means that guen a 8 > 0 there exists some nonempty open set, Ui, such that 10. The Daniel! Extension 185 n(U s ) < S. Since U s is nonempty there must exist some element g € Us- By the right invariance of n then we have that n(Us) = But, since e 6 Uig~ l , we have established the existence of an open set, 0 = C/jjr 1 , contain- ing e such that n(O) < <5. Now since one can show that C 0 (<7) is dense in Li(G ) , given any £ > 0 there must be a function/ 6 C 0 (G) such that f | m - f | d,i < .f. J a But f \m\dn < f \m - f\d/i + f \f\dfi J a J a J a and in general, for an arbitrary measurable set E, [ |/| dn < max/ • n(E). J E Hence, since we are unrestricted in the choice of S, it is clear that we can choose an open set 0, containing e, such that / | m | d/i < £ ■’o for any £ > 0. With t given and an appropriate 0 chosen we now select a symmetric neighborhood of e, V, such that V 1 C O.f Denoting the characteristic function of V by kv we have k v (x) = (kv * m) (x) = / kv(xy- l )m(y) dv(y) J a for a.e. x € V. t The existence of such a V is guaranteed by observations 10 and 11 in Chapter 6. 10. The Daniel) Extension 187 3. Let 1(f) = fa f(x) dp be a right invariant integral on G. Let J(f) = faf{yx) dp. Show that J is also a right invariant integral, and, hence, J(f) = A(y)I(f). A (y) is called the modular function of G. Prove that A (y) is a continuous homomorphism of G into R. 4. Using the notation of exercise 3, show that if G is compact or commutative, then A (y) = 1 for all y 6 G. 5. Let I be an integral on Co(X) , where X is a locally com- pact Hausdorff space. If / € C<>(X ) , define ||/|| = supxex |/(x) |. If E is any compact subset of X prove that there exists a constant k = k( E) such that 1(f) < fc |[/(| for all / € Co(X ) provided [x |/(x) ^ OJ C E. References J. Naimark, Normed Rings. 2. Loomis, An Introduction to Abstract Harmonic Analysis. 3. Rudin, Fourier Analysis on Groups. CHAPTER 11 Characters and the Dual Group of a Locally Compact, Abelian, Topological Group In this chapter we shall continue the discussion of the Banach algebra L}{G) that we introduced in the last chapter Next we shall turn our attention toward the concept of a character and of the dual group of a locally compact, abelian topological group Some examples of characters in familiar frameworks and certain theorems pertaining to characters and the dual group are then discussed Remark It is easily verified that Li(0) is an algebra with involution if we take for/ 6 Li((?),/*(x) = f(xr l ) Throughout this chapter we shall assume that G, in addi- tion to being a locally compact topological group is also abelian As noted in the last chapter L } (G ) is an algebra with identity if and only if G is discrete In any case by the results of Chapter G the Banach algebra Li(G) can be ex- tended to an algebra with identity which we shall denote by R(G) hence /?((?) is a commutative Banach algebra with identity which we shall denote by e f To facilitate matters in the discussion that follows we introduce the following notation Let / € Li(G) \\ e now define/* y) to be My) = /(v*- 1 ) We now contend that the mapping X -»/, is uniformly continuous 1 HcVe toaX we to toe xised Wi Ctiaptasa S *®d. 4 insofar as the multiplicative identity of the Banach algebra I* being denoted by e 188 1 1 . Characters and the Dual Group 189 Proof. Let / € Li(G) and let e > 0 be given. Since C 0 (<?) is dense in Li(G) there exists a function g £ C 0 (G) such that 11/ — g l|i < e/4. Since g € C 0 (G), however, this implies that there exists a compact set, E, such that g(CE) = 0 and that g is uniformly continuous by the first theorem in Chapter 7. Hence there must exist some neighborhood of the identity of G, V, such that \g(y ) - aiyx- 1 ) | 6 < 4p(£) for yiyx- 1 )- 1 or 1 ff(v) • - g*(y) 1 € < 4 y(E) for x 6 V. Consider now • II g - g* Hi = / 1 J G g(y) - g*(y) 1 dn(y) Since g x (y) vanishes if yx~ l $ E we can write II g - g x !li = [ I g(y ) - 9*{y) I J E\jEx If x € V, we have which implies 11/ - /- Hi < 11/ - g lit + il g - g* Hi + II fc - U Hi < 7 + T + 7 = e for x e V '^ 4 4 4 ( 1 ) t Note here that the right invariance of the integral implies that 11/ -Plli - |l/. -fcllt. 190 Eementi of AbitrotS Homcnc Anctyuj We now claim that /* -/» • /-Ar. (2) because (ft - /•) (z) =/»(*) -f,(z) - /(z^ 1 ) - /(fir 1 ) while (/-UW = (/-A-K*r*) »/(£T J ) -fizz-hy-*) « /(er~«) - which demonstrates that their effect on any group element, z, 13 the same and proves their equahtj In light of this ii/.-/, ii. = ii(/-/^)«n. which, by the nght invariance of the integral, is equal to 'I (f - fm) Hi But, as eh<3wn by (!), 11/ - /*r* Hi < * t or ipr x € 1 , or, using (2), HA - /. Hi < < for yr->0 (3) which is the *tatement of uniform continuity and completes the proof Our next theorem is concerned with an approximate identity for Lj{G) (See Chapter 1, p 9, where the notion of approximate identity is fir^t mentioned ) Theorem 1. Let / £ Lt{G) and let «> 0 be gnen Then there exists a neighborhood of the identity of G, I , *uch that if ii is a nonnegatne Borel function which vanishes outs’de V, with the added propertj that f u(z) dn(z) = 1, J G then |j/ — / »tt |ji < < (u is an approximate identity) 1 1. Characters and the Dual Group 191 Proof. By the preceding result, with x taken to be the identity element of G in (3) , we can assert the existence of a neighborhood, V, of the identity of G such that 11/ — /v (fi <« for y e V. (4) Let u satisfy the hypothesis and consider ( f*u)(x ) -J{x) = f ifixy- 1 ) - f(x))u(y) dyiy). J G We now have ||/*M-/|li= [ dy(x) \ [ if(xy~ l ) - f(x))u(y) dy(y) J G I J G < [ dyix) [ | fixrr 1 ) - Six ) | • I uiy) \ dy{y) while interchanging the order of integration (which is justified by the final answer) yields I! / *u ~ / ||i < J I uiy) | dyiy ) J ! /„(*) - fix) | dyix) or ll/*«-/||i< f uiy) || / v - fWidviy). J G But, since u vanishes outside V , the last term is equal to f uiy) || / v - fWidyiy). J v Using (4) this must be less than « / uiy) dyiy) J v which, using the hypothesis about u, is equal to just e or ||/ *u -/||i < « which completes the proof. 1 92 Elements of Abstract Harmonic Analysis This next theorem has a certain esthetic appeal as well as being a rather useful result about any homomorphism mapping a commutative Banach algebra with identity into the complex numbers Theorem 2 Let X be a commutative Banach algebra with identity, e, where it is assumed that the norm of e is unity, and let / X — ► C be a nontrivial (therefore onto) homo- morphism Then JJ/|{ = 1 Proof Let x € X and let f(x) = X Since / is nontrivial we immediately have /(e) = 1 , hence /(* - Xe) = /(*) - X/(e) «= X - X = 0 which implies (x — Xe) 6 M, where M is the kernel of / Since / is a homomorphism onto C, AT must be a maximal ideal Therefore, by the first theorem in Chapter 4, x — Xe must be singular which is equivalent to saving that X € <r(x) By part (7) of the last theorem in Chapter 3 we have IM < II *11 <=*!/(*) I < 11*11 for any x € A which implies ii/ii<i m By the definition of [(/ j| we have !/(*) I <11/1111*11 for all zZX so, letting x = e, l - !/(*)! <11/11 II e II *■ 11/1) Combining this with (G) completes the proof Corollary Under the same hypothesis as the theorem, / must be continuous Characters and fhe Dual Group 11 e now proceed to the notion of a character of a group a notion that will play an important role throughout the rest of this chapter and the next chapter 1 1. Characters and the Dual Group 193 Definition. A mapping x: G->C with the properties that 1. | x(s) | = 1 for all x £ G, and 2. x(zy) = x(x)x(y) is called a character of G. We will denote by r the class of all continuous characters on G. If we take (xixi)(ff) = xi(ff)xs(ff) as the law of composition, it is easily verified that r is a group and is called the dual group of G. Denoting the exten- sion of L\{G) to a commutative Banach algebra with identity, e, by R(G) we wish now to determine the set of maximal ideals in R(G). First we shall demonstrate that Li(G) is a maximal ideal in R(G), but before proceeding we ask the reader to recall that a typical element of R(G) is ae + f where a is a complex number and / £ Li(G). With this in mind if we suppose that I is an ideal that Contains Li((?) we note that this would imply the existence of an element ae + /, a ^ 0, which is in I but not in Li{G) . Since / is an ideal, which implies {ae + /) — / = ae £ 1 e £ I and, hence, I = R(G). Therefore Li(G), which is clearly an ideal, is a maximal ideal in R{G). Having gotten this one maximal ideal we will obtain the rest through the follow- ing theorem. Theorem 3. Let x be any continuous character of G, take / £ Li(G), and consider fix) = Jf(x)x(x) dp(x). 194 Element* of Abslroct Hormonic Anolyju Then the mapping R(G)~>C x fixed, Xe + / — ► X + J(x) !> + / varying is a homomorphism of R(G) onto C that does not vanish on all of Li(G) Conversely, every homomorphism that docs not vanish on all of Li(G) is obtained in this fashion Further- more distinct characters induce distinct homomorphisms Remark Theorem 3 establishes a one-to-one correspondence between the dual group, r, and the class of all homomor- phisms that do not vanish identically on Li{G) Since there exists a one-to-one correspondence between the class of maximal ideals of an algebra and the set of all homo- morphisms t the theorem also establishes a one-to-one corre- spondence between the maximals ideals excluding Lt(Q) and the dual group Proof J et x 6 r and consider the mapping of 11(0) onto C mentioned in the statement of the theorem It is immedi- ately clear that the mapping is a linear functional In light of this we shall now demonstrate that the mapping preserves products to establish the fact that it is a homomorphism To this end consider the product! (Xe + /) (^e + ff) ~ Xpe + Xy + ft/ +f*3 which, under the above mentioned mapping, maps into Xu + Xp(x) + m?(x) + /(/ •g)(y)x(y) d**(y) •’o t See m this connection the mapping mentioned in Chapter 3 1 » the theorem following the introduction of the quotient algebra t The reader should clarify for himself exactly what is meant by tins product of elements in R[G) 1 1 . Characters and the Dual Group 195 which is the same as V + \g(x) + /i. f(x) + [ x(y) dfi(y) f f(yzr 1 )g(z) d/x(z). J Q J G (•) Interchanging the order of integration in the last term and noting that, since x is a character, x(y) = xfzjxO/z"" 1 ) , the above integral becomes f x(s)g(z) dy(z) [ }{.yz- l )x{yz-') dy(y). J G J G But, since the integral is right-invariant, ( *) becomes Xp + \g(x) + yf(x) + §(x)f(x) - (X + f(x))(y + g(x)) which, since we have already noted that the mapping is linear, establishes the homomorphism. We must now show that there is at least one / € Li(G) that does not map into zero under the above mapping. To this end take k £ Li(G) such that [ h dn 0 . •’a As the required / we can now take / = h/ x it being clear that we are never dividing by zero because x is a character. Hence we have now shown that the homo- morphism does not vanish identically on L\{G) and, also, by the same argument, shown that the homomorphism is nontrivial and, therefore, onto. We can now proceed to showing that, given any homomorphism, there exists some character that “generates” it in the fashion illustrated above. Suppose now that h is a homomorphism mapping R(G) onto C that does not vanish identically on Li(G), and con- sider the restriction of h to just L\{G) ; i.e., take hi = h It, ((;>• 196 Elements of Abstract Harmonic Analysis By the preceding theorem we have that |j A || *= I which implies 0 < || Ai |{ < 1 Now, since G is a locally compact, abelian, topological group, the dual space of £i(<7) is iso- morphic to t and, since || Ai)| < I, one can, by using a well-known representation theorem, write hi (/) - M*) for some $ 6 L„{G) Furtherznorcuehave || |L = |1 h j) < 1 Bearing these results in mind Jet/, g E Li(G) and consider / m /)*(*)♦(») My) - Mf) f g(y)My) Mv) - hi(/)Ai(ff) (7) which, since /, g € Liifi) and h x is a homomorphism on L\(G) t must equal M/'ff) = /(/•(>)(*)*(*) Mx) - / ♦(*) Mx) { f(xy~')9(y) My) t Suppose X is a set S is a nng of Bets in A' and measure on S Now let £ £ S In this context L m (E, u) *» L m (E) consists of those functions / such that |/(z) | < M ae on E Denoting the class of all M that bound |/(r) | a e by M * it can be shown that the esimtud tup off, inf M — css eup/ ~ |1/||« i» actually a norm on the space L m {E) Furthermore it can also be shown that for/ € L m {B) toll/ll, - 1«4 / ~ *UP/ r-® / It ia assumed of course, that p(E) < « in the above discussion 1 1. Characters and the Dual Group 197 Interchanging the order of integration this becomes Jg(y) dfi(y) j 'P{x)f{xy~ l ) dfi(x) = J g(y) dfi(y) J 4'(x)f v (x) dfi(x) = f 9(y)hi(f v ) dp(y) or, combining the first and last terms of this equality, / h U)g(y)'Ky) dy(y) = J hi f v )g(y) dn(y) G J G for any g 6 Li(G), which implies hi(f)Hy) = hi{f v ) a.e. Now, by the first result in this chapter, / is a continuous function of y. By the corollary to the second theorem to this chapter, hi is a continuous function and we can now say that the composite function hi(f v ) is continuous. Since h, by hypothesis, does not vanish on all Li(G'), we can choose an / £ Li(G) such that fh( f) ^ 0 and can assert that the function 'f'iy) = hUA hi(f) is continuous almost everywhere. We can now define /(y) to be continuous everywhere because this will not disturb our representation theorem [namely, that h(f) = jf{x)4>{x) dn Or)]; i-e., we will take tty) = hi(fv) hi(f) (8) 1 98 Elements of Abstract Harmonic Analysis everywhere Using (8) and noting that = (/,)» we have - m/w> = «(/,).) - h(f.My) - hUlHxmv) and can conclude that *(*y) - tK*Wy) (9) As noted previously || p [[„ < I which implies !*(*)!< 1 ae (10) Since p is continuous however if | p I > 1 at x a then there must be a w hole neighborhood b of r» such that \p \ > l for all * € U Since n(U) must be positiv e this contradicts (10) Hence !*(*) I < 1 everywhere From (8) we have the fact that tK*i) « I where ei is the identity of G Using this and (9) we have for any x Combining this result with (10) gives | p(x) | = I for all x Thus p is a character on G and we have expressed h t (/) in the required way It now remains to show that h can be expressed in this fashion Jvote that in the previous notation we now have MS) -SiP) I^et Xe + / € f?(G) and consider *(**+/> ->*<«> + MS) Since h is a homomorphism fc(e) *= 1 and we have +/) - * + /«■) which is the required representation for h 1 1. Characters and the Dual Group 199 To show that distinct characters induce indistinct homo- morphisms let xi and xi be distinct characters from r, and suppose that, for any / £ Li(G), X + /(xi) = X +/(»). This implies f f(x)xi(x) dfi(x) = ff(x)xi(x) dn(x) J G J G for any / £ Li((7). Hence Xi = X 2 a.e. If two continuous functions are equal almost everywhere, however, they must be equal everywhere, which implies that Xi = X2 and contradicts the assumption that xi and * 2 were distinct characters, and proves that distinct characters induce distinct homomorphisms. QED Having established the 1-1 correspondence cited in the remark and noting the 1-1 correspondence between the class of all homomorphisms and the maximal ideals, we wish to rewrite some of the results obtained here in terms of the notation used in Chapter 4. Let h be a complex homomorphism on R(G) and let M be the corresponding maximal ideal. Then, in the notation of Chapter 4 we have for (Xe +/) £ R{G) H\e + f) = (Xe + /) (M) = X +f(M) — X + hi( f) = x + fix) = X + f f(x)x(x) d»{x). (ID 200 Element* of Abstract Harmon e Analysis Similarly we can rewrite (8) as x(*) MM W) ’ f.W) S(M) for / such that h t {f) 7 * 0 (/(M) * 0) (12) We would now like to show that any x 6 T in addition to being continuous must also be uniformly continuous and the notation used above will be helpful in proving this Theorem 4 Let x be a continuous character of G Then x 1 3 uniformly continuous Proof Consider I x(i) - x{«) I i/.w - um 1 l/<M) 1 B> the last theorem in Chapter 3 we have ~/,WI < II/, -A Hi iron 1 “ i/wi It now remains on)> to apply the first result obtained in this chapter to complete the proof Examples of Characters Example 1 I-et G be the additive topological group of real numbers It 4- and consider the conditions imparl on any character x namely that for ti tt (: R x(h + k) = X(ti) + XM and I x(0 1 = 1 for t e R A solution is offered by taking x(f) ** t i,r where x 6 R and thus a character is obtained for each x € R Actually, these exhaust all continuous characters of R Now consider 1 1. Characters and the Dual Group 201 Li(G) and its extension, R(G). Using the notation of Chapter 4 we obtained Eq. (11) which, in this case, becomes \+f(M) = X + nf(t)e«*dl. — co Since x is characterized by x we can write KM) = /(x) =f(x) = fme^di * —CD where x is the x corresponding to x- Example 2. Let G be the additive group of integers (with the discrete topology) , Z, +. Since this is a discrete structure Li(G) has an identity present and we need not bother with extending it. We wish now to obtain solutions to x(«i + rw) = x(«i) + x(« 2 ) («i, E 2) and ! x( n ) | = 1 for all n 6 Z. Analogous to the preceding result we note that x(n) = e' na (a € R ) does the trick. We now let/ € Li(G) and proceed in the exact same fashion as the preceeding examples to get [using Eq. (11) againj KM) — fix) = f,Kn)e *- n~~ca and note the 1-1 correspondence { X } e ia ( a mod 2 tt) where (x) is the class of all characters. We wish now to focus our attention on the topological aspects of r. Suppose M represents the set of all maximal ideals in R{G) and let Mo denote the maximal ideal Li(G). We have then the 1-1 correspondence M - {Mo} «-*■ r. 202 Element* of Abstract Harmonic Analysis With respect to the topology defined in Chapter 5 for Af it was shown that Af was Hausdorff and also compact (see first theorem in Chapter 6) Since, in general, removing one point from a compact Hausdorff space j lelds a locallj com- pact space (considers g , removing one point from a closed in- terval) we have that Af — JAfo} is a locally compact, Haus- dorff space We will topologize r now by taking the open sets to be those which correspond to open sets in A/ — |A/«) under the 1-1 correspondence and, with respect to this topology, can immediately assert that r is a locally compact, Hausdorff space To indicate more exactly what the corre- sponding topology in T is, we will exhibit a particular member of the fundamental system of neighborhoods in r To this end let xi be some particular element of T, let Afi he the corresponding element in A? — {Af#}, and consider the following n elements of R(G) gt = 4- /*, A, » 1, 2, •••, n Then the neighborhood about xi, denoted by W(xufi ■**» t) is given by all those x such that |<fc(A/) - | - \MAI) -M*h)\ = | Jfk(x){ x (x) - xi(i))*(x)| <« for k = 1,2, •••,» (Note that in the above expression the Af that appears in the inequality is the Af that corresponds to the particular x ) We shall presently prove that T is actually a topological group Since it has already been demonstrated that r is a group and that r is a Hausdorff space, it remains only to verify that the mappings and X -* X“‘ (x € r) (xi» xj) — * Xtx* (xi. xi € r) are continuous 203 11. Characters and the Dual Group „ .i i, we have obtained a topology for r, as it happens there is a slightly nicer characterization and this next theorem is needed in obtaining this topology. Theorem 5. The mapping G X r — > C (x, x) x(z) is continuous. Proof. Consider Mi ^ Mo- Now take (for / 6 U (G)) | / xl (Mi) - /*,(M) 1 < ~ 1 + \f xi (Mi) - fxi(M) |. By the last theorem in Chapter 3 [part (7)] we have \f xl (Mi) - fx t {Mi) 1 < \l/-i " which, by the first result in this chapter, can be made a rbi- trarily small for 6 W (for some neighborhood of the identity IF), whereas | /..(MO - /»W I . can be made arbitrarily small, less than . say, ^ “"“j M to 7 (Mi /„, «). [See Chapter 5 for the definition ox V{M h f xv e).] We have now established the continui y the mapping „ G X M - {Mo} -> C (x, M) —*fx(M) where /is a fixed element of Li(G) '• v that It is easily seen that there is some / € i ( /(Mr) X 0. At any point (xi, Mi), then, /(M) ^ °/r ,^//(M) is a con " neighborhood of Mi-t Consequently M argument, tinuous function of x and M by the preceding argu t The reader is asked to recall now that the top <=> ^ functions in Chapter 5 was the weakest topology that would make x(M), x fixed, M varying, continuous. 204 Elements of Abstract Harmon e Analys This m turn implies bj Eq (12) that the function of x and a: x(*) X(*) MV) fon is continuous It is to be noted that given any M there is a corresponding x and this is the x that is meant above This completes the proof W e now w i«»h to determine another basis for r s topologj It is first noted that denoting the fundamental stem of neighborhoods on r by | II ) since the 11 form a fundamental system of neighborhoods for r that if we can construct another sj stem of sets JA ) such that any 11 contains some % then clearly the JAT) form a fundamental system of neighborhoods for T also Let E be a compact set in G xo € I* t > 0 be given and take A'Xxo B t) - (xllx(x) - xi(z) I < « for all* € E\ It is first observed that this set is open by a theorem in Chapter 5 t where for the sake of comparison with that theorem we have GXT->R (x x) -* i x(x) - xo(x) I We will now show that the sets A (xo E «) form a funtla mental system of neighborhoods for r Before proceeding we note that for any /* £ Lj[G) there exists a pt € Co(G) such that L I/* - Pt I d/t < tj 4 t The theorem nmndis if A X > — »- ZOO 0 open then the set |ylMx v) € O for alt* € b £ compact! 1 1. Characters and the Dual Group 205 Since there must be a compact set, E k , such that p k (CE k ) = {0}, we have Assuming n such /* have been given we will denote the com- pact set n u Ek by E. fc— X Now take S < min , k = 1,2, •••,: x 2 H/* H* ' and suppose X € N(x o> E, 5) where x £ We now have ]//*(*) (x(s) ~ xo(x)) dn(x) < / !/*(*) I I x(x) ~'Xo(x) I dp(x) J o < J i fh(x) I | x(x) — Xo(x) | dft(x) + f \ft{x) | | xW — Xo(x) I dy.(x) Jg-E Hence < 6 i| f k ill + -J < e for k = 1, 2, •••, n. N(x o, E, S ) C TF(xo,/i, •*•,/»>«) which completes the proof. 206 Element* of Abstract Harmonic Analysis Exercises 1 Prove that if G is discrete r is compact, and if G is com pact T is discrete 2 Let X and 1 be commutative Banach algebras with identities and let K be senusimple Show that if / is a homomorphism of X into Y, then / is continuous 3 Let m ms and h, be Bore! measures on a locally compact Hausdorff space X such that = hi + « Prove that the regularity of any two of them implies that of the third 4 Let G be a locally compact topological abelian group Show that G is the union of a disjoint collection of <r- compact sets |F«} and each <r-compact subset of G is contained in the union of a countable subcollection of the U?,j X is called a-compact if X = F, compact References } Naim&rk Normed Ring) 2 Rurhn Fourier Analyst* on Groups CHAPTER 12 Generalization of the Fourier Transform to LifGj and L 2 (G) In this chapter we will define the generalization of the Fourier transform; a result toward which our previous work has been directed. First we shall define the Fourier transform of any function in L t (G ) where G is a locally compact, abelian, topological group, ( G will carry this connotation throughout the rest of this chapter) and will then briefly sketch the Fourier-Stieltjes transform after some necessary notions about complex measures have been introduced. In addition to briefly discussing some peripheral material that arises, the Fourier transform over Li(G) will then be ex- tended to L 2 (G). The Fourier Transform on Ii(G) Having shown V to be a group and having defined a topology on T, we now wish to show that T is a topological group. Hence it must be shown that the operations of taking inverses and that of multiplication are continuous. First we will show that multiplication is continuous and to this end suppose N(xox'o> E, <0 is a neighborhood of xox'o- We must now find neighborhoods of xo and xo> Nj and N:, such that NiN 2 C N(xox' 0 , E, e) and we contend that N( X o, E, e/2) and N (xj, E, e/2) are suitable neighborhoods. To demonstrate the desired inclu- sion let X 6 N( x o, E, e/2) x' € W( X J, E, e/2) 20 7 and 208 Elements of Ab itroct Harmonic Analy* $ and consider txGrix'Cy) - x»(y)xi(y) t = lx(y)x'(y) - xo(y)x'(y) + xo(y)x'(y) - xo(y)xJ(y) 1 - Ix'(y)(x(y) - x»(y)) + (x'(y) - xJ(y))xo(y) I £ I x'(y) 1 1 x(y) - x<i(y) | + 1 x'(y) - xj(y) 1 1 xa(y) I < i + \ ** * for a11 v e E This implies xx' € -^(xjxJ, B, *) and proves that multiplication is continuous To show that taking inverses is continuous it is first noted that x(y)x(y) - 1 for any y 6 G implies that (x(y)>-‘ - x-Hy> - x(v) or x -1 “ x Now given a neighborhood of xo” 1 . N()&\ B, t ), we must find a neighborhood of xo which is mapped into it We claim that N{xt>, E, *) does the trick, for suppose x £ N(xo. E, «) Then I x~'(y) - xrHy) I - I x(y) - xo(y) I = I x(y) - xa(y) l * lx(y) - xo(y) I < * which implies x -1 6 N(xt\ E, t) Having proved that r is a locally compact, abelian, top- ological group v.e can proceed to define the Fourier transform off € L\(G) as fix) = Jf(y)xiv) d»{y) (i) 1 2. Generalization of the Fourier Transform 209 In view of the 1-1 correspondence between r and the maximal ideals, M [excluding Li(G ) ], if M £ M and "corresponds” to x we have fix) = f(M). (la) Further, since we threw the weak topology on r, the above functions are each continuous functions of their respective arguments.! Some immediate consequences of (1) or (la) are, for fi, h € L\(G) and X £ C, / \ + / 2 )(x) = fi(x) + Mx) (2) /\ m (x) = x/i(x) (3) /\ fi*Mx) — /i(x)/s(x) - (4) In Chapter 6 the notion of an algebra with involution was defined. In the case of L\(G) if we take f*(y) = /Or 1 ) as the involution operation, it can easily be verified as noted on p. 188 that it is an algebra with involution. We now have /*(x) = ff(y~ 1 )x(y)d l i(y). G being abelian, however, implies that, for any measurable Be t, E, n(E) = n(E ~ l ) . Replacing y by y~ x in the integrand of the above expression, we obtain /*(x) = ff(y)x(T 1 ) dn{y). •>a But, since x 6 T, xClT 1 ) = x -1 (f) = x(y)', hence f*(x) = 7(x) t See footnote to Theorem 5, Chapter 11, p. 203. 212 Elements of Abstract Harmonic Analyui It can be shonn (cf Halmos [S3) that { m I is a measure With this result in mind we define p to be regular if | p J is regular (See p 215 for the definition of regularity ) Wen ill denote by B( Y) all those complex, regular, Bore! measures, p, such that ImI(K) < « Defining addition and scalar multiplication of the set func- tions in B(Y) bj 1 m Pi € B(Y), E € 5, (#»i + =» pi(E) + Pi{E) and 2 A 6 C p£ B(Y), E e S, (\p)(E)=\ p(E) we now claim that B{ Y) is a Banach space with respect to the norm !MI = Utm p£B(Y) Some additional measure theoretic results about complex measures mil non be stated after making the following Definition A signed measure is a countably additive, ex- tended real-valued set function p on the class S (in general, on a ff-rmg of subsets of an abstract set) such that n(0) =0 and such that p assumes at most one of the values or Remark 1 (Jordan decomposition theorem) t Let P € BO) Then p can be written as P = Px ~ Pi + t(n» — P*) where the p. are measures and each p, £ BO) We can restate the result of this theorem by asserting that t One al?o has such a decomposition if n « ju«t a complex, finite- valued measure , in which ease the corresponding measures p, finite-valued (not necessarily regular) 12. Generalization of the Fourier Transform 213 i can be written as = X + tv where X and „ are signed measures which belong ioB{Y). For ix a complex measure one defines jjd, - jf dpi - + '//*" "" 'll*"' Another import! consequence o! the above theorem^ that even when complex measures P ”, g0 define d. plement the theorem and dea wi turn our Having defined ”'on calsian products attention toward comple. , we we re dealing just as we spoke of product measures when we w with measures. . „ c \ Remark 2. Suppose we have two "-sure and (F, Sr) and let m € B{X), v 6 B(Y). for all A 6 Sx and all B € Sr, (uXr)(AXB) ~»{A)v{B). c v c / gee Chapter 10) is then The extension of y. X * to S x X r{ and „ are the desired measure. It can be shown that, regular, then so is y X v. nreceding work, having Again, in close analogy wi measura ble sets in the defined a complex measure on double integrals, product space, we wish now + be Fubini theorems, iterated integrals and new versio y. If now Remark 3. Let / be a Borel function on X X „ 6 B(X),r 6 B{Y) and one of f [ 1/1 d\ Jx J Y jj^f\ d\y\ d\v\, f xxy \ fl is finite, then f Mux, i-IJf'-Uf** JXX.Y Element! of Abitract Harmonic Anolyih 214 We wish now to show that after having introduced a suitable multiplication that H(G) is a Banach algel ra To this end let n >€ B(G) and let h bo a measurable set in 0 Wc now take tm " {(x v) € GX(7|*y € Since /■<* must be a Bore! setf ) nG X G the follow ingdefini tion makes sense wo define for any measurable set ]• (n'»)(#)-(nX»)(J«») (7) Tins done we now make the follow ing Contortion 11(G) with respect to the operations and norri already indicated is a Banach algebra There arc now a number of things that we must v enfj in order to prove this and first we will show that ft •»* is a countably additive set function Countable Add tiv ty ltd y r £ B{G) and let F - VF where the F are pairwise disjoint and measurable Now wc can write („.»)(/•) - (, •»)(u I ) - (f X')(«(/ ).) which since n X v is a complex product measure is equal to fi(i*x»)(f >, - £(*•>)<* > ami cstal lisbes the countable additivity of a • v Next we will show that a • v is regular f To venty that Ft > » meomrat leoccp 1S1 wbm con* dwd the i «tM>rc of / • e 1 2. Generalization of the Fourier Transform 215 Regularity. All that will be shown here is that the conditions for inner regularity are satisfied; the argument for outer regularity following in a similar fashion, f We also may as- sume, making use of the Jordan Decomposition Theorem, that ii and v are, in addition, measures. Suppose first that E is a Borel set and K is any compact set contained in E. We now have (/* * v) ( K ) <<jx*v) (E) which implies sup(/x » r) (if) < ( ii*u)(E ) (incompact). (8) KCE If we can now show that for any e > 0 there exists a compact set, H, such that H C E and {n *v)(H) > (m *v)(E) - e, this will imply that (p *v)(E) is the least upper bound of the set ( (m * v) ( K ) | K C E, K compact} and we can change the inequality in (8) to an equality. To this end we note, by Remark 2 above, that n and v being regular implies that ( n X v) is regular. Thus for a set E m , where E is measurable, and e > 0, there must exist a compact set F C E m such that ill X v){F) > (|iX v)(E ( 2) ) — e. t It is recalled here, that if a Borel measure, />, satisfies the condition that for any Borel set, E, (a) ii(E) = sup n(F) (F compact), F*ZE then it is said to be an inner regular Borel measure. If a Borel measure ft satisfies the condition that, for any Borel set, E, (b) n(E) = inf /t{0) (0 open). ECO Then ft is said to be an outer regular Borel measure. A Borel measure that satisfies both (a) and (b) is said to be a regular Borel measure. We note that every Haar measure is regular. 216 Dementi of Abjtroct Hcrmonic Analyut Now consider the continuous mapping p GX G — G (x, y) — » xy and let // denote the image of F under this mapping Since F C E<», then p(F) “ // C P(£«>) C E or IIC E and, since the mapping is continuous, // must be compact In addition we ha\e p -, (//) C Hm But p~'(//) contains F so F C //«> Using these facts we now have (» ")<//> - (u X *)(//«,) > („X»)(f) >(nX ►)(£<«) - « * in»p)(E) - * which completes the proof Having shown p • v to be regular we must now show that it has a finite norm Then we will have the fact that (p •>>) € B(G) To show that B(G) is a Banach algebra however, we must also show that || n • » |( < |( v || ■ || v || One notes immediately that if this inequality holds and |i ► ( B(G) that this will certainly imply that (n • r) has a finite norm for any /j, v 6 B(G) In light of this vve will pursue proving only the norm inequality, knowing that we will simultane- ously complete the proof that B{G) is closed with respect to the operation • Proof that H a • * || < H u || * |1 r || I>ct p, »• € B(G) and consider (/*•*)(£)“ f **’) where, as usual, we reserve the symbol of k with a subscript to denote the characteristic function of the set represented 1 2. Generalization of the Fourier Transform 217 by the subscript. We also have, now, (m * v) ( E ) = (ft X v) ( Ey >) = f k BM (x, y) d(/j X r). J GXG Since ( x , y) £ E m <=* xy £ £we can rewrite the last integral as (#* X v)(E m ) = f k E (x y) d(fi X v). ■'GXG Since | m | (G) and | v | (G) are finite we can apply Remark 3 of this chapter to justify writing the last integral as an iterated integral to get fk E d(n*v ) = f f k E (xy) dy(x) dv(y) . (9) Since (9) holds for any characteristic function it will cer- tainly also hold for any linear combination of characteristic functions. By defining a linear combination of characteristic functions to be a simple function we can restate this last result by saying that (9) holds for any simple function. Thus, using (9), we have for any simple function,/, [ f f(zy) dii(x) dv(y) = ffd(n*y). (10) -"ff -Iff •’g We now wish to avail ourselves of the following two theorems. Theorem. For any complex, bounded, Borel measurable function, /, there exists a sequence of simple functions {g„| converging uniformly to / and, conversely, every uniformly convergent sequence of simple functions converges to a bounded, Borel measurable function. Theorem. If a sequence of functions converges uniformly on a set such that the total variation of the (complex) meas- ure of the set is finite, then the operations of limit and inte- gration can be interchanged when integrating over this set. 21 B Element* of Abstract Harmonic Analysts In \ien of the«e two theorems v>c can assert the validity of (10) for c\ery bounded, Borcl measurable function,/, the interchange on the left being justified bj noting that | n \ (G) and | v j ((?) are finite and being justified on the right bj the following equal ltj (m*v)(C) - (m X f) ((»<«) = (aXO(CXG) = M (C)»(0), and recalling the definition of the integral with respect to a complex measure Bearing these things in mind, non consider IIm'HI - U'H(G) which, by exercise 1 of this chapter, is equal to sup [fd(» •») i/is« I J o 1 Using (10) we now ha\e II * II “ sup I f f f(zy) dn(z) dt(y) I l/ISi I J o J a I < sup f f | f(xy) I d |n | (x) d \ v | (y) i/isi Jo Jo £11 *11 II *11 which completes the proof Jflaxmg shown that B(G) is a Banach algebra wc wish to demonstrate now that it is a Banach algebra with identity and we claim that the identity clement is gnen b> |1, e 6 EC G to<E) = [0 tU where t is the identity element of G taxing as an exercise for the student the \enfication that a, fc B{G) we shad demonstrate here that a. does ha\c the required property namclj (hat for anj n £ B(G) ^ * fit <U> 1 2. Generalization of the Fourier Transform 219 In view of the Jordan Decomposition Theorem (Remark 1 of this chapter) we need only prove that (11) is satisfied for the case when p is a measure. In light of this let p he a meas- ure from B(G ) and consider (m *He)(E) = (p X He)(E ( ») = j Ve((E( 2 )) x ) dfl(x) where (E m ) x denotes an z-section of E ( 2 >.f We now note that (E m )x = {y I (X, y ) 6 E m \ = {y | xy 6 E ) = {y I V € xr l E] = x~'E. Hence the last integral becomes f fi e (x~ l E) dn(x) Jg which, by the definition of p e is just f d M {x) = y{E). QED J E The Fourier-Stieltjes Transform Using the same notation as in the preceding discussion let X € T. We will now associate with every p £ B(G) a func- tion jionr via £(x) = [ x(y) dfi{y) Jo t See the discussion of product measures in Chapter 10 for the defini- tion of an z-section and the justification of the equality written above. 220 Elements of Abstract Harmonic Analysis and define this to be the Founer-Shctijes transform of p We will now examine what this means in the framework of the real axis Example 3 Let G = R, + Then, as noted earlier in this lecture, due to the one-to-one correspondence between the continuous characters of G and the elements of G we can write the rouner-Stidtjcs transform of p 6 B{G) as Hx) - (x ( J!> wlicre the integration is to be performed m the Lebesgue- Sticltjcs sense and a(<) is of bounded \anation on the real axis This situation occurs because in tins framework anj p € B(lt) is induced b> a function of bounded variation on the real axis We now wish to extend the Fourier transform on fii((7) to Lt(G)f but must first introduce some preliminary notions Positive Definite Functions Definition Let // be a group and let <p be a complex-valued function on ll <p is said to be positne definite if J2 X^^y.^') > 0 for an> Vi yr, • • yn 6 // and Xi X, • ■ X\ € C Example 4 Using the same notation as before, if / 6 Li((?) then the function v> « /' •/ is positne definite and continuous f Li(G) denotes the Banach spare of square inferrable function* over <7 with norm of / € U(G) defined as where n is a fjiven Haar measure on <7 1 2. Generalization of the Fourier Transform 221 The proof that ip is positive definite is straightforward and is left as an exercise (see exercise 3 at the end of this chapter) . We prove that <p is continuous. Let/, g £ L 2 ((?) . Since C c (G) is dense in L 2 (<?) there exist sequences, {/„} and {g„ } , of functions in Co((? ) such that ll/»-/li*-*0 and || ffn - g || 2 -» 0. Now we have I (f*g)(x) - if n *gn)(x) | < f | S{xy-')g{y) - fn(xy~ l )g*{y) I dy(y) which equals f I (/(sjr 1 ) -/» (xy~'))g(y) J G + (g(v) - 9n{y))f n {xy- x ) | dy(y) < ||/„ - /||» || g || s + 11 9n — g Hz II fn II 2 using Holder’s Inequality and, since ||/„ || 2 is bounded, we see that fn *g n -»/ *g uniformly. Moreover, denoting f x (y) — /(yx -1 ) , as in the beginning of Chapter 11, we have I ( fn * {In) (x) - (fn* ffn) (z) | < f l/n(zr') -hizy- 1 ) II 9n(y) I Mv) -'<7 < || (/„)*-. - (/„)*-> ||i max | g n (y) |. yzG But || ( ff) x-t — {fn)z - 1 ||i can be made arbitrarily small for xsr 1 belonging to a suitable neighborhood of e, V, by our first result in Chapter 11. Thus, fn * is continuous for each n, and, consequently / * g is continuous. Applying this result to the functions/, f* £ L 2 (G) yields the desired result. 222 Elements of Abstract Harmonic Analysis The definition of a positiv e definite function immediately implies the following three statements (1) If e is the identity of G and v is positive definite, then *>(«) > 0 (2) Let <p be positive definite and let y £ G Then v>iy~ l ) - v(y) (3) Let $ lie positive definite let y be anj element of G, and let e be the identity of G Then I v>(y) I < s?(e) The first statement will be proved here to illustrate how the proofs of these statements go the proofs of (2) and (3) follow in a similar fashion and ore left as an exercise for the reader Proof of (1) All we need do to prove this statement is take iV = 1 j/i = e and As *= J in the definition of a positive definite function Example 5 As before fcl T denote the dual group of G and let f» be a measure m #(T) ic n > 0 Then v(y) = J x(y) <Mx) is continuous and positive definite, namely £ <Wx) ■ .-i J r • nt-i .x(*») J dp(x) > o recalling that xW) = x(*-) 1 - x(i-) Thus v is positiv e definite To prov e that v> is continuous w e firvt note that if E is a compact «et of F then the set K{ r« E, t) = \x j| x(*) - x(*«) I < « for all x € / } 12. Generalization of the Fourier Transform 223 is an open set in G. The proof is analogous to that, given after Theorem 5, of showing N(x o, E, e) is open in T. Next we note that I <p(y ) \ < [ \ x(y) I dfi(x) = f cfc(x) = m(t) = j|.^ |j. «' r -'r Now, since p is regular, there exists a compact set E C r such that n(CE) = n(T) - p(E) < 5 where S is a preascribed positive number. Thus, I v(x) — \ < I \ x(x) ~ x(zo) | dn(x) Jo - / ! x(z) - x(zo) | dp(x) J E + f ix(z) — x(xo) I dp(x) •>CE < e I! M !| + 26 for all x £ N (x 0 , E, t) which proves the continuity of <p. The following theorem gives us a complete characteriza- tion of continuous positive definite functions in the general case. Theorem ( Bochner ). If <p is a continuous function on G, then <p is positive definite if and only if there exists a p £ B(T), P > 0, such that for all y € G 224 Element! of Abjtroct Hormonlt Analyvi is positive definite and continuous The remaining half the proof vv ill be giv en in the appendix to this chapter Thus in the ca=c of the addtttv e group of real numb every positive definite continuous function on It can be i tamed by the following liebcsgue-Sticltjes integral v(y) - f c "daft) where a(() is a monotone increasing function of bourn variation on the real axis and com er«ol> In the case of the real axis however Bochners thcor can actually be made stronger insofar 03 we no longer requ that <p be continuous but only measurable i e we have i following result On the real axis if v is positive definite a measurable then there exists a monotone increasing fund: of bounded variation on It a(t) such that ?(«/) *= J « *da(l) ac Denoting the class of all positive definite functions on by P and the class of all linear combinations of members the «et Li(0) n P by £Lj{G) n P] we can now state t following theorem Theorem A [/^(G) n P~\ is dense in L\(G) and L\(G) Proof t\e first observe that if / g € C e (G) then f *g Co(G) In particular we show that if A is the compact si port of / and B the compact support of g then the supp< of / • g is contained in AB For since / » 0 on CA and g *■ on CB f(ry~ l )e(y) - o except if y € B and ry~ f A or z (. Ay which in plies t! x € AB Consequently /• g v am hes on the complement the compact «ct AB T1 erefore / • p € C#(G) bv virtue of t proof for Example 4 1 2. Generalization of the Fourier Transform 225 It follows now from Theorem 1 of Chapter 11 that the set of all f* * g, where f,g £ C 0 (G ) , is dense in C 0 (<?) , and, there- fore, dense in Li(G) since C 0 (G) is dense in Li(G). However, /* *g = i(/ + g)* * (/ + g) - l(f - g)* * (/ - g) + ~(f 4- fff)* * (/ + ig) - \U ~ w)* * (/ - ig). But each of the functions (/± g)* * (/± g), {f± ig)* * (f ± ig) is positive definite and continuous. Therefore, f* *g e [TrfG) n p] and consequently, C^i(^) n ^3 ' s dense in L\{G) . The fact that [Li((?) n P] is dense in L 2 (G) is proved in a similar fashion using the fact that G^(G) is dense in L«(G) . Now, just as in the first two chapters, having defined the Fourier transform over Li(G) we now wish to focus our attention on the problem of recovering the given function from its transform. First let us consider the family A of all functions of the form/, where / 6 Li((?). We recall from our initial discus- sion that / is a continuous function on T. Moreover, Eqs. (2)-(5) of this chapter show that A is an algebra of func- tions, ivith respect to the customary operations, which is such that it contains the conjugate of any of its functions. In addition, since /(x) = f(M) , where M is the associated maximal ideal with x, the final theorem of Chapter 3 shows that the algebra separates points, i.e., if xi X; there exists an / 6 A such that/(xi) ^ /(xz)- Next, given x with associ- ated M, M 7^ Li(G ) , there exists an / 6 Li(G) such that /f I; whence, /(M) ^ 0, by the previously quoted theorem. In other words, for each x € r there exists an / £ A such that/(x) 5 ^ 0. Finally, since the one-point compactification 226 Element! of Abstract Harmonic Anatym of T amounts to adjoining M 9 *= L\{G) to M — j V,} and •since for / € Li(G), f(M t ) = 0 we «ee that each / € A vanishes at infinity Hence on the basis of the extended Stone-A\ cierstrass theorem (see appendix to this chapter), A is dense m the Banach space of all complex \a!ued continu- ous functions on r which \amsh at infinity, where the norm on the space is the usual supremum norm (\\ e recall in gen era! that a real or complex valued function defined on a locally compact IlausdorfT space is asid to rantsA of infinity if for c\ cry « > 0 there exists a compact set E C. \ such that |/(x) l < t for all * £ CE) \\ e apply these results to establish the follow mg Theorem B Let »• € B(T) If Jx(v) Mx) - 0 for all y € G then ► = 0 Proof Lct/€ Li(G) Then Jf(x) Mx) = J'Jfiv)x(y) My) Mx) - Jf(y) dM(y)fx(y) Mx) - 0 Thus jt(x) Mx) - 0 for any / € L\{G) Since by our previous discus, ion the family A of all / is dense in the space of all continuous func- tions on r which i amsh at infinity we clearly ha\ e that Js(x) Mx) - 0 1 2. Generalization of the Fourier Transform 227 where g is any continuous function on r which vanishes at infinity, but the mapping T(g) = f g(x) d\v\( x ) Jr is a bounded linear functional on the Banach space of con- tinuous functions on r which vanish at infinity, since I T(g) | < f | g(x) I d | v\ (x) J r < ii mu ii where || g ||, as noted, is sup* £ r | ff(x) I- Now since T is bounded and T(g) =0 for all g in the Banach space, it follows from the Riesz representation theorem (see Appendix to this chapter) that v = 0. With this result in mind we can now proceed to the inver- sion formula. Theorem (inversion formula). Let / £ £Li (G) n P~\. Then fix) € Lj(r) and, if the Haar measure p of G is fixed, the Haar measure of T, p, can be normalized so that f(y) = J f(x)x(y) dp(x)- Proof. We note, first of all, on the basis of Bochner’s theorem that / can be written in the form f(y) = f x(y) Mx) J r where v £ B(T). Now let g £ Li(G). Then (ff*/)(e) = j g(y~ l )f(y) dp(y). J G ( 12 ) 226 of Abvtrort Kamenk Anonym But ft (x) *Cx) - j f fj Cy)x(y) <fe(y) tfr(*) * fs( sr‘) <fe(y) / x(y) d>( x ) “ fv(v~ , )f( y) <fo(y). Consequently (»•/)<») -/«x)*(x) (13) Let A € [£a((7) n P] and apply this re«ult to the function y • A € Li(G) Then r ((*•*)•/)(<> - ( ff*A)(x)*(x) J T “ J fffx)J(x) *(x) by Eq (4) of this chapter On the other hand, (fp-A) */) (e) - ((p-/)-A)(e) and since h € [Li((7) 0 ^3* another application of (13) implies that «s •/)•*> W- fe l*)/(x) ■&(*) where X € B{I*) is «uch that Mv) - Jails' ■»(*> (») 1 2. Generalization of the Fourier Transform 229 Comparing these results, we see that J g(x)Hx) dv(x) = J g(x)fix) dx(x) (15) for all g where g € L\(G). Since, as observed earlier this family is dense in the Banach space of all continuous func- tions on T vanishing at infinity, ( 15) holds for all such func- tions, and an application of the Riesz theorem shows that Hx) dv{ x ) = fix ) d^(x)- (16) Next, let w € C 0 (r) and let E be the compact support of w. By our observations preceding Theorem B, we know that there exists a«( L\ (T) such that v(xo) ^ 0, where xo € E. However, Co(r) is dense in Li(r), and, therefore, for arbi- trary e > 0, there exists a u € C 0 (r) suc h that II f - «l|i < «• But- recalling the last theorem of Chapter 3, we have I Hxo) - «(xo) | < || v — u ||i < e. It is, consequently, clear that there exists z 0 6 Co(r) such that Zo(xo) t 6 0. Furthermore, by the result following Eq. (5) of this chapter /\ z* * Zo(xo) = I Zo(xo) I 2 > 0. /\ /\ Since z* * z 0 is continuous, z* * z 0 is positive in some neigh- borhood, xoUo, of xo, where t/ 0 is a suitable neighborhood of the identity. Now, to each x« 6 E we associate such a z a 6 C 0 (r) and such a neighborhood U a of the identity. Clearly EC U XaU a a and since E is compact E C U It is then clear that, if h = z* * Zi + • • • + z* » z n , then h > 0 on E. Moreover, recalling the proof of Theorem A, we have that h 6 C'o(r), and by Example 4 ,h £ P, and, therefore to ELi((7) n P]. 230 Elementj of Abifroet Harmonic Anatyjii He now define the mapping F C*(T) — * C b)f F(tc) = ( vh~ l d\(x) J r where X is characterized \ia Eq (14) We first note that F 13 well defined for if / 6 [Xi(G) n P} and if / > 0 on F, Jwfi' l dX(x) *= j wfi-'j-'J d\(x) - fvt'J-'l, *(x> = ( tr/ -1 «Mx) J T using Eq (1G) C!carl> F is a linear functional on C%{ T) Now, since h is a continuous positive definite function ne ha\c by Bochncr’s Theorem and Theorem E that the associ- ated X of Eq (14) must be a measure, whence, if tr > 0 F{ic) > 0 Next it is clear that one con find a tr € C»( F) and a suitable / £ £Li(G) n P~] «a> / £ Li(G) D P fo that the associated » of Eq (12) is a measure, and such that / lr(x) <Mx> X s 0 Hence tr/ £ C«(r), and F(trJ) = fxr/h-'d\(x) ** /it <M x) J* 0 J r m F 0 t In the discussion below k~ l and/* 1 devoir 1/k and J// 1 2. Generalization of the Fourier Transform 231 Finally, let w £ C 0 (T) and let xo € r. Let h be as above, i.e. h > 0 on E, the compact support of w. If we let f(y) = xo (y)h(y), then fix) = / h(y)xo{y)x(y) d/i(y) J o while h(xox) = I Hy) (xox) (y) dn(y) J G = f h(y)xa(.y)x(y) dn(y) so f(x) = h(x ox) - Therefore, by Eq. (16) Hx) dvix) = hix ox) dX(x) or Hx) dv(x) = h( x ) dXixo'x) I whence r(E) = X(xJ‘ 1 E). Thus setting v(x) = u>ixo l x) we have F(v) = fw( x ^x)ihix))- l Mx) •'r = [ w(x)ih{xox)) 1 d\(xo l x) J r = [w(x)ifix))- 1 dHxo 1 x) = [ wix)if(x))' 1 dvix) J v 232 Dementi of AbjfroCf Harmonic Analyst* Recalling the discussion of Haar measure in Chapters 8 and 9 in particular A Sltasure Theoretic approach in Chapter 9 ivc have that F(») * / te(x) dp(x) w here p is a Haar measure on r Let / be any function now in [_Li(G) n PJ and Jet u> € Co(T) with h as above then if * is related to / by (12) fw rfr(x) “ ftefh 1 rfX(x) J r J r “ F( tc/) “ / «/dp(x) J r Since this is true for any tc 6 Co(T) wc can now conclude that dr ^ f dp Butr € B{T) so |/ r /Jr| S^<M-IMI<« Hence ) € £j(r) Finally /(») - £t(v) <Mx) “ J}{x)x(y) dp(x) QED As an immediate consequence of tins theorem applj mg it to the case of the real axis viewed as an additive group we have m - />- ■<" Comparing this to the inversion formula obtained in 1 2. Generalization of the Fourier Transform 233 Chapter 1 we immediately note that the factor 1/2? r does not appear here because we have normalized the Haar measure (Lebesgue measure in this case ) ; i.e. we have re- placed fi by (h/2tt) where y represents just Lebesgue measure. At the end of Chapter 11, we showed that the collection of all sets of the form N (xo, E, e), where xo <E r, e > 0 and E is a compact set in G constitute a fundamental system of neighborhoods for F. In the course of establishing the con- tention made in Example 5 of this chapter, we proved that the analogous sets N (x 0 , E, e) , where x 0 £ G, e > 0, and E is now a compact set of T are open. With the aid of the inver- sion theorem, we can now establish that these sets form a fundamental system for G. Namely, let V be an arbitrary neighborhood of e. There exists a neighborhood U of e such that UU* 1 C V. Moreover, by exercise 4 of Chapter 6, there exists a neighborhood W of e such that W C U, and, since G is locally compact, we may assume that F = W is compact. Then FF~' = WW~> C UU~' C V. If we denote by k? the characteristic function of the set F, and let / = fcp/ju(F) I/2 , then f* */, using Example 4, is con- tinuous and positive definite. Furthermore, (/**/) (x) = [ dn{y) J G = f dii(y) ( 17 ) J G = “r f k F (yx- l )kp(y) dp.(y). m(F) J g Now, k F ( y) = 1 for y £ F, and k F (yx~ l ) = 1 for yx~ l 6 F or x € F~ l y. Consequently h = /* */ vanishes on the comple- ment of FF~ l , a compact set. Therefore, h € C 0 (G) fl P C Li(G ) 0 P, and, by the inversion theorem, we have that h € Li(T), and fHx)x(y) d P ( X ) = Hy). J r 234 Element! of Abstract Hormone Analysis In particular, jHx) dp(x) - h(e) - (/* */)(0 * —rz: [ b(y) d?(y) pin J c « 1 by Eqx (17) We also observe that , /\ , h =/*•/= 1 / 1' >0 Now it is a simple matter to show (<ee exercise 7 of this chapter) that the set function * defined bj r(S) - J (h(x) Mx) l or S a Bore] subset of T is a regular measure, whence, there easts a compact sub-et E of T «uch that r(CE) < t where we take * to be a given positive number less than one Thus I *= w(E) + p(CE ) < r(H) + « so / Hx) dp(x) > 1 — « J E However, Hy) = f hxdp + f hx dp, J s J cx and if y € .V (e, E, *) |x(y) - 1 1 <« for x € E, so or I — Rex(y) < Rex(p) > 1 2. Generalization of the Fourier Transform 235 Thus I h(y) | > \ f hx dp — ( hdp \ j e I J CE > (1 - e) f hdp - v(CE ) E > (1 - e) 2 ~ e- If we take, for example, e = then ! h(y) 1 > 0 so y 6 FF~ l C V, and, consequently, N(e, E, J) C V. Thus the open sets N(e, E, t) form a basis at e; hence the open sets XoN (e, E,e) = N (x 0 , E, e) form a basis for G. Next, we observe the following important fact: if x 0 € G and Xo 5 ^ e, then there exists a x 6 T such that x(xo) 5^ 1. Namely, since x 0 ^ e, we can find a neighborhood V of e such that xo £ V. Hence, using the above notation, there exists an N ( e , E, «) C V, so x 0 £ N ( e , U, «) , i.e. | x(zo) — 1 | > e for some x £ E; whence x(zo) 1, and the contention has been established. An equivalent statement to the preceding is clearly the following: if x lt x 2 € G and x 2 ^ xo, then there exists a x € r such that xfci) ^ x(* 2 ) ■ The next theorem will essentially provide the way by which we can extend the Fourier transform to L 2 (G). The Fourier Transform on L 2 (G) Theorem (Plancherel) . The mapping Li«?) n L 2 (G) -*■ Lz(r) /->/ is an isometry onto a dense subspace of L 2 (r) and, hence, can be extended to an isometry of L 2 (G ) onto L 2 (T). 234 Element* of Abstract Harmonic Anofyiii Proof We shall first show that the mapping h an isometry by showing that, if / € U. n U ||/iii - \\j\\ t or, to para- phrase this that the mapping preserves norm?) To this end let/ € Iq n L, and consider g = /* •/ By \irtiic of Example 4 of this chapter it follows that g € P, hence g € Lin P But ll/ll! - J - //(•©« - //(*)/*(*-') A*(*) - s(') (18) where t is the identity of G By the inversion formula of this chapter however, we can also write g(e) as g (e) - fo(x)x(*) rfp(x) * f fir(x) dp(x) (10) J T But /\ Hx) - (/• */)(x) - I/M I’ so that, substituting this m (19) wc obtain ff(<) - ( I /(x)Na(x) - 11/Hi J r Combining this with (18) then completes the proof that the mapping is an isometry Assuming the v ahdity of the rest of the theorem wc can now extend the Touricr transform to Lt(G) as follows Since Li(G) fl Lt(G) is dense in Ly(G), for any / € L,(<7) there exists a sequence of functions | /.| , from ln{Q) fl ln{G), such that n in 1 2. Generalization of the Fourier Transform 237 { /„} is, therefore, a Cauchy sequence. By the isometry just demonstrated, the sequence { /„ J must be a Cauchy sequence also. Since L 2 (T) is a Banach space we can say that fn — *•/ 6 L 2 (T) and it is / that we shall define to be the Fourier transform of f £ Li. Having made this definition we must show that it is a sensible one; i.e. that/ is well defined. To this end suppose ii in /»— >f and H H, h n — * f where /„, /i„ € In fl L 2 for all n. Since { /„) and { A„ } each converge to / in the 2-norm, then 11 fn — K U 2 — > 0 or h n — >/. The “ontoness” of the extended mapping is seen in a similar fashion. Thus it just remains to show that the given mapping, /— »/, where / 6 Li((?) n L 2 (G) is a mapping onto a dense subspace of L 2 ( F) . We denote by A the image space of this linear transformation. Suppose / 6 A; consider for any x £ G the function g(x ) = x(x)f(x)- Define h{y) = f x (y) — f{yx~ l ). Then clearly h (E Li(G ) fl Li(G) and . ft(x) = f(y*~*)x(y) dy(y) J G = J f(y)x(yx)-dn(y) = xO) Jf(y)x(y) dp-iv) = x(*)/(x) = ff(x). Bemenh of Abttraet Harmonic Analytii 23fl i e g «* h Therefore, A is clored under multiplication by xOO Now, let V be anj function in Ia(T) and let tr ~ / be an} function in A Suppose (p, tc) = / ptP dp ■» 0 J T where {<p, w) denotes the customarj inner product in the Hilbert space Z/»(r) It follows b> what we just proied that /*(x)®(x)x(*) dp(x) “ 0 (20) for any x € O By Holder’s inequality ynfc 6 Li( F) , hence the set function, defined for alt Borcl sets, E, of r by r(E) - / v(x)Mx)x(*) Mx) belongs to /i(T) (*ec exercise 7 at the end of this chapter) But, for arbitrary y € G fx(v)Mx) - /n>(x)ti(x)x(xy) Mx) -0 b} (20) , whence using Theorem B we have that r ■* 0, or / *(x)tf(x)x(;r) dp(x) - 0 1 B for every Borel set h of T which clearl} implies that ■* 0 ae Moreover, if we set y(x) “ /(xx«) where/ <E A and x* € r, then the function h(y) - /(y)x«( v) € li(G) n E*(G) and J(x) ** ^/(y)x»(y)x(y) d/r(y) “ /(xx») - 0(x) Thus, g € A Hence suppose/ € Li(G) n /’C A(G) 0 Li{G) 1 2. Generalization of the Fourier Transform 239 and / 0, then/ 6 A, and / ^ 0 by the inversion theorem, say /(xo) 5^ 0. If we let xt be an arbitrary element of r, then defining w by w(x) = fixxT'x o) we have by the above that w £ A and w(xi) = f(x o) ^ 0, Consequently, since the elements of A are continuous, w does not vanish in some neighborhood. However <pw = 0 a.e. for all w £ A. Thus, we must have clearly that <p = 0 a.e. Recalling that in Li(<?) we identify functions which are equal almost everywhere this implies that <p — 0; whence 0 is the only element of L 2 (r) orthogonal to all of A. We now employ the following result from Hilbert space theory (cf. the text by Halmos [[4]) : If M is a closed sub- space of a Hilbert space X, then M = where in general the orthogonal complement of M, is defined as [y 6 X | (x, y) =0 for all x 6 M). It follows directly from this that if M is just a subspace of X, then M = M x± . In our case, with X = L 2 {T) and A = M, we have A = = (Or = L 2 (T) which completes the proof of the theorem. On the basis of the Plancherel theorem, we can show that the algebra, A, considered prior to Theorem B, of all func- tions /, / € Li(G) is just the set of all vi * <P 2 where <pi, <pt 6 L 2 (r). For, by Parseval's formula (see exercise 10 of this chapter) with g in place of g, we have that, for /, g Cz L 2 (G) J f(y)g(y) dy(y) = Jf(x)g(x ') <Wx)- ( 21 ) Next, substitute g(y)xa(y) f° r {/(!/) bi (21). This yields j f(y)g(y)xo(y) dg(y) = jf(x)g(x ox -1 ) <Wx) (22) = (f*g)(x o) 240 Hementi of Abitroct Harmonic Analyst! /\ where ire haie used the fact that ffx*(x~ l ) " f(x*x~*) ^ctr since every element h 6 Lj(G) is of the form fg where/, 9 £ L*(G), we have b> Eq ( 22 ) that h(* # ) “ (/*p)(x«) for arbitrary x« € I\ or h « / * g but / g £ Lj(r) by Tlan- cherel s theorem Conversely, t! f g € Lj(r), then ( 22 ) show s that / * g belongs to the algebra A since f • g - fg, and fg € Li(G) We can now establish the follow mg Theorem C If B i* 0 is an open set in r, then there exists an / £ A [where /t is the algebra of all / / 6 Li(G)3 such that / * 0 on CD, and / ^ 0 Proof Let E be a compact subset of R such that w ith p as in the inversion theorem p(E) >0 E exists by the regu* lanty of the Haar measure p and the fact that fi is positive on any nonempty open set By property 12 of Chapter C there exists a neighborhood 1 of e which we may assume compact since T is locally compact such that \ E C R Let k r and A* be the characteristic functions of l and E, respectively A y kg € Lj(r) since I and E arc compact Moreover k r »kg « / € A by the preceding discussion and (Av*A,)(x # ) «= j kvixoX ’)kt(x) <*p(x) But Arfxrt"') “ 0 if x«X ‘ C 1 and Mx> - 0 if x $ f / - 0 on C(1 E) Z> CR Finally jhx> Mx) - f f M*) dp(x) - /l.W)*«)/wrf ’>■«*) - Mf>(l ) > 0, whence, / p* 0 Be now wish to mate a statement about the dual group of T Let y be a fixed element of G and define, for x £ I\ f,(x) - x(y) 12. Generalization of the Fourier Transform 241 We first note, by Theorem 5 of Chapter 11, that f y ( x ) is continuous on r. Next we note !/v(x) I = 1 for all x, (23) and, for xi, xs 6 V Mxixz) = (xixsM y) = xi(y)xi(y) = /v(xi)/u(x 2 )- (24) In view of this we can say that /„ is a character on r. Now, denoting the dual group of r by A, we can say that for each y € G there is some f y 6 A. Thus we have the following mapping: G —> A V-*fv In this case we also note that ym ~>fvJ V s because ( ftnvi) (x) = x(yi Vi) = xiyOxM ~ fvi(x)fvt(x) = ( fvifvz) (x) • Hence the mapping (24) is a homomorphism. But we can actually make a stronger statement about this mapping and will do so in the following duality theorem due to Pontrjagin. Theorem ( Pontrjagin) . The mapping (7 — > A y ~*fv, given above, is an isomorphism and a homeomorphism. Proof. We first observe that the mapping is one-to-one, for if y h y 2 £ G and yi ^ y 2 , then, by the discussion following the inversion theorem, we know that there exists a x 6 P such that x(Vi) X ( 2 / 2)1 i- e - ^ /«• Thus the mapping mentioned in the statement of the theorem which we will denote by F is an isomorphism of G into A. 242 Elements of Abjtrort Harmonic Analyili Next, we show that F is a homoomorphism of G onto F{G) Let E be a compact «et in r and consider the sets of the form V — A r (i* E , «) where i* € G, « > 0, and V — A r (tfo, E, e) where 6 A The sets U constitute a fundamental system of neighbor- hoods for G bj the discussion immediately following the in\ ersion theorem w htle the sets V form a fundamental sy stein about x» by the discussion at the end of Chapter 11, and if V =« N(F(x o), E «), then f(i/)-rnF(G), (25) namely if x € V then | x(*) - x(*a) l < « for all x € F, or l/.(x) — /« ,(x) | <« for all x € E winch means that F(x) -/,(! ft F(G ) Conversely, if/, € F(C) n V, then |/,(x) - /n(x) I < * for all X € A, which implies that x € V and /, =* F(x)tF(l/) Now (25) clearly implies, since «e aircad} Inon that F is one-to-one that F i« a homoomorphism of G onto F(G) To fini«h the proof therefore, we must ju«t show that F(G) ■= A We first show that F(G) is a closed subset of A \\c already know by the preceding that F(G) is a locally compact subspace of A If we let F(G) * and A* be the one- point compactifications of F(G) and A, rcspectit cly (see Chapter 7 where the one-point compactification is discussed), F(G)* may be considered as a subspace of A* since any compact set in F(G) is also compact m A But F(G)* is closed in A* by Theorem 4 of Chapter 5 Thus, F(G)* fl \ “ F(G) is closed in A If w c can now show that F(G) is dense in A then F(G) »■ b (G) ■» A, and ire will ha t e com pleted the proof If F{G) is not dense then F((7) “ b ((») 9* A and CF(G) r* w open Hence, by Theorem C, there exists an K £ A, where here, os u«ual /I designates those } such that / £ Ai(F), such that h H 0 and h *• 0 on F(G) Now 1 2. Generalization of the Fourier Transform 243 where h £ Li(r), and <p £ A. However, h(F(x )) = 0 for any x £ G. Thus, J h(x)x(z) dp(x ) = J h(x)fz(x) dp(x ) = 0 (27) for any x £ G since 0 = k(F(x)) = fh(x)Mx) dp( x ). J r Now applying Theorem B to Eq. (27), similarly to the last part of the proof of the Planeherel theorem, here taking v{E) = / fik(x) dp(x) for an arbitrary Borel set E of T, we can conclude that h — 0 a.e. Therefore, h = 0, which is a con- tradiction, and the proof has been completed. We shall just draw one consequence from this duality theorem; namely, it follows immediately that the dual theorem to Theorem B is true: if p £ B(G) and if p(x) = Jox(y) dp(y) = 0 for all x € T, then p = 0. The duality theorem has a great number of further useful consequences and we refer the interested reader to the references for some of them. Exercises 1. Prove that j| p || = | p | (G) = supi/|< 1 1 fof dp | where / is, of course, Borel measurable. 2. If <p is a positive defin ite function. on G, prove that, for x £ G, <p(x~ l ) = <p{x), and J p(x) | < <p(e). 3. If / 6 L 2 ( G) , show that tp — f* */ is positive definite. 4. Let f be a positive definite function on the locally com- pact group G. Show that if <p is continuous at the identity element, e, then <p is uniformly continuous on G. 5. If H is a closed subgroup of G then prove that the dual group of G/H is isomorphic and homeomorphic to the subgroup of T consisting of those characters which are constant on H and its cosets. 6. If p, v, X £ B(G), prove that (p * v) * X = p * (v * X) and p * y = v * p. 244 Element* ef Abitrocl Harmonic Analyil* 7 Let X 1)0 a locally compact H&usdorft space and n be a regular Borel measure on A' Define the sot function r as follows ,(£) - f f(x) dn(x) where/ € Li(X,m) (see Appendix) Prove that v £ B(X) 8 Let X be a locally compact Hausdorff space If P(A') designates the set of all continuous functions on A' which vanish at infinttj , then show that C 0 (A') is dense in P(A’) with respect to the supremum norm 9 Using the same notation as m exercise 8, show that / € 1 (X) if and only if tho continuous extension of /, which we again denote b> /, to A’*, the one point com- pactification of A is such that f(x*) « 0, w here x* is the adjoined point (see Chapter 7) 10 Ut / g € Lt ((?) Prove Parscval’8 formula Jf(y)0iV) <Ml/) ” jf(x)o(x) dp(x) Appendix to Chapter 12 W e list here a number of results and theorems relevant to Chapter 12 and also present a proof of Bochner’s theorem Ut A be a locally compact Ilausdorff space W c designate bj V(A') the Banach algebra of all complex-valued, con- tinuous functions on A' which vanish at infinity and with supremum norm Theorem {extended Stone-}] etertlrass theorem). Suppose A u a Eubalgebra of V(.Y), such that 1 / 6 A implies / € A 2 If x: Jj € A' and x x r* x\ then there exists an / € A such that /( ti) t* f(xi) 3 If x» € A, there exists an / € A such that /(x«) r* 0 Then A is dense in 1 (A ) 1 2. Generalization of the Fourier Transform 245 Riesz Representation Theorem. Let / : V ( X ) — > C be a bounded linear functional. Then there exists a unique p £ B(X) such that f(y) = [ y(i) dy{t) (i) J X for y € V ( X ) . Moreover |j n || =- 1| / jj where H / 1| designates the norm of the bounded linear functional /. The converse of this theorem is also true and quite simple to prove; i.e., if M € B(X), then (1) defines a bounded linear functional on V(X). If X is a locally compact Hausdorff space and p a regular measure on X we let Li(X, it) designate all those Borel functions, /, on X such that !l/lli = / 1/14* is finite. For n regular it can be shown that Co(X) is dense in Li(X, it) with respect to the norm || ||i- Similarly one defines L„(X, m), 1 < P < °°, and has the corresponding statements. If v € B(X), we say that v is absolutely con- tinuous with respect to it if v{E) =0 for any Borel set E such that it(E) = 0. It is easy to see, that for / G Li(X, ju), the set function v, defined for each Borel set E as follows: v(E) = has the properties that it belongs to B(X) and is absolutely continuous with respect to p. The converse also holds: Radon-Nilcodym Theorem. If v € B(X) and is absolutely continuous with respect to the. measure n, then there exists an / G Li(X, p.) such that, for any Borel set E of X, 246 Bem*ntj of Abjtrocl HormorJc AnoIyj3» One frequcntfj writes f *= drfdn, or dr « fdn and speaks of the Radon-N ikodym dematnes {see, for example, Halmos £5])) Proof of bexhner's Theorem. Let be continuous and posi- tive definite W e maj assume that j* 0, and consequently that <r (e) = 1 recalling properties 1 and 3 of a positi\e definite function mentioned in the chapter It also follows that ^ is uniformly continuous on G bj exercise 4 of this chapter Next, choose / 6 C#(G) The function is uniformly continuous on EXE where E is the compact support of / Consequently , recalling the definition of the integral there exist disjoint Bore) sets E„ t « J, 2 •••, n such that l l^ x E ( «= E and such that 2 /(*.)/(*>) *»(x.*7‘)i*( E,)n{E t ) i mi where x, (• E„ approximates / f/(*)7(v)v>(*y~ l ) du(x) duty) ( 2 ) J a J o as clo«ely as desired The finite sum is certainly nonnegative since p is positne definite, whence (2) is nonnegatne for any / € C«(G), but C»(G) is dense in L\(G, ft) - L\(G) so, for any / £ Lj(G) , this is also true V, c now define F L\(G) C as follows f(S) - { f (*)*(*) dn(x) ■'o F is clearly linear, and |F(/)I< [\f(v)\\*(v)\Mv) J Q < *uplf»(y) tll/lli (3) tt fit. 1 2. Generalization of the Fourier Transform 247 since <p(e) = 1. Now, for/, g £ Li(G), define U,Q) = = f (/*?*) (x)<f>(x) dy(x) J G = J j f{xy~ 2 )g{y~ 2 ) v (x) dy(x ) dy(y) = / f(x)g(?r 1 )‘p(yz) <Mz) = J ff(x)g(y)<p(xir l ) dy(x) dix(y). It now follows from this last relation that ( /, g) is linear in the first argument and conjugate linear in the second argu- ment, and, as we already know (/./) = F(f */*) = J J f(x)f(y)<p(xy~ 1 ) dix(x) dy(y) > 0. G J G (/, Q) just misses being an inner product since (/, /) = 0 does not necessarily imply that / = 0. But the properties that (/, g ) possesses are already sufficient to establish the Cauchy-Schwarz inequality. Thus I (f,g) | 2 < (/,/)(<?, ?)• Since <p is uniformly continuous there exists a neighborhood V of e such that I <p( xr 1 ) ~ <p(x) i < e and I <p(xi r 1 ) - 1 1 < t for y £ V and xy~ l £ V. Let U be a compact symmetric neighborhood of e such that U 2 C V. Then for x,y £ V, x y~ l £ U 2 C V and, of course, y € V. If we let g — ku/y(U ) , where kv is the characteristic function of U, then g € Li(G) and 248 Ekmeti ti of Abt fracf Harmonic Anofy»ft one has (/,<?)- ?(/) m fjl*) J W***) -•?(*)) Mv) M*)> and («,g) — l «* ' fo(7/))« J 0 ~ l Ma<x)rfa<y) It follows that | (/, ff) — F(f) | and | (g, g) - I ( can he made arbitrarily 'mall, but |F(/> I* S !*•(/) - </,»>)* <1^(/) - (/,?) I* + </./)(?. tf) < I f</) - (/. If) |* +</,/) I (f f) - I I + (/,/), w hence |F(/)!*< (/./) - **(/•/•> (4) for an j / € J,,(G) Now let ft = / • /•, so ft* «= ft, and defino ft* •» ft*' 1 • ft*' 1 for n *» 2, 3, We may apply (4) successively to the A*, n => I, 2, •••, since each of these functions u in L\{G) Doing this we get i r<n i * < m < < ... < (F(a>-)),- < where the last inequality follows from (3) But hm H A*" 11*"' - r,(A) by the results of Chapter 3, where r,(A) we recall is the spectral radius of A Thus, \nf)f<r.W - supM(.V) I 12. Generalization of the Fourier Transform 249 by Theorem 3 of Chapter 4, but sup 1 h(M) | = sup | h( x ) | MeM xeT since h vanishes at infinity, i.e. h(Mo) = 0 for ikf 0 = Li(G ). /\ However, h(x) = / */* (x) = |/(x) I 2 by the consequence of (5) of Chapter 12. Consequently, | F(f) 1 < sup|/(x) |. (5) x«r We can now define a mapping, which we again denote by F, on the set A of all / such that / € Ln(G) into C, by F(f) = F{f). The mapping is well defined, for if / = g, (5) immediately implies that F(f) = F{g). The functional F(f) is of coiurse linear and is bounded in the supremum norm by (5). By the Hahn-Banaeh Theorem (cf. Naimark pj) F can be extended to a bounded linear functional on F(T) , the space of continuous complex-valued functions on I' with supremum norm, such that the norm of the extended functional is the same as that of F. We again denote the extended functional by F. An application now of the Riesz Theorem listed earlier in this appendix shows that there exists a v 6 B{T) such that F(g) = fff(x -1 )*(x) for all g€ F(r), * / r and imi = imi < i by (5). In particular F(f) = F(f) = J fix' 1 ) dv( x ) = j fix) d)i{x) jx{x) dvix). 250 Efemonli of Abifraet Harmonic Anotyili Hence, Ft/) - (h !)*)«») - //(i) d M (x) /i« Mx) for any / € L%(G) eo *(*) - / x (i) *( x ) •'r for a c i € G, but both *»{*) and J7x(x) fMx) arc continu- ous on G, whence *(*) ** fx(x) *(x) for all x € G Finally, 1 « v>(e) ~ / rf-(x) - r(D < IMt < t, / r therefore »>(?) =■ ||f|| which readily implies that f is a measure on T and completes the proof of the theorem R» TEnENCW 1 N nmsrlc Wormed Rirgt 2 Hudin Founer Analyst on Group* 3 Loomt* An Introduction to Abtlracl Harmonic Anahjtu 4 H&Imoe Introduction to Hdbcrt Spate 5 Hal mew M rann Theory Bibliography Bourbaki, N., Topologie Generate, Book III, Chapter I. HernSann, Paris, 1951. Goldberg, R. R., Fourier Transforms. Cambridge Univ. Press, London and New York, 1961. Halmos, P. R., Introduction to Hilbert Space. Chelsea, New York, 1951. Halmos, P. R., Measure Theory. Van Nostrand, Princeton, New Jersey, 1950. Hewitt, E., and Ross, K., Abstract Harmonic Analysis, Vol. 1. Academic Press, New York, 1963. Kelley, J., General Topology. Van Nostrand, Princeton, New Jersey, 1955. Kilmogorov, A., and Fomin, A., Functional Analysis. Vol. 1: “Elements of the Theory of Functions and Functional Analysis.” Graylock Press, Rochester, New York, 1957. Loomis, L., An Introduction to Abstract Harmonic Analysis. Van Nostrand, Princeton, New Jersey, 1953. Naimark, M., Normed Rings. Nordhoff, Groningen, 1959. Gel'fand, I., Raikov, D., and Silov, G., Commutative Normed Rings (Amer. Math. Soc. Transl. No. 5), Amer. Math. Soc., 1957. Natanson, I., Theory of Functions of a Real Variable. Ungar, New York, 1955. Pontrjagin, L., Topological Groups. Princeton Univ. Press, Princeton, New Jersey, 1946. Rudin, W., Fourier Analysis on Groups. Wiley (Interscience), New York, 1962. Taylor, A. E., An Introduction to Functional Analysis. Wiley, New York, 1958. Van der Waerden, B., Modem Algebra. Ungar, New York, 1949. Zaanen, A., Linear Analysis. Wiley (Interscience), New York, 1953. 251 ate continuity of a measure, !45 utely convergent trigono- netric series, space of, 26 rence point, 62 >ra mach, 25 mpletely symmetric Banach, 210 sets, 125 algebra, 125 ith involution (star, sym- metric), 97, 188 ytic vector-valued function, 29 roximate identity, 9, 190 inch algebra, 25 inch space, 2 is t a point, 73 if open sets, 73 ontinuous mapping, 62 chner’s Theorem, 223, 246 rel ;omplex — measure, 211 inner regular — measure; outer regular — measure, 215 measure, 126 regular — measure, 215 sets, 125 mnded linear functional, 29 inonical mapping, 110 artesian product (of topological spaces), 78-79, 85 lauchy Integral Formula, 32 Integral Theorem, 31 lauchy-Hadamard Formula, 32 lauchy-Schwarz Inequality, 17 character, 193, 200 closed set, 62 closure, 62 cluster point, 123 compact, 81 locally, 85 compactification one-point, 118 content, 163 inner, 164 regular, 164 continuity uniform, 107 continuous mapping, 62 partition of the identity, 121 convergence in a topological space, 123 of a generalized sequence, 116 convolution, 6, 180 coset, 50 covering, 81 finite, 81 open, 81 countability first axiom of, 73 second axiom of, 74 countably additive class, 125 Daniell extension, 158, 166 directed set, 116 discrete topology, 61 dominated convergence theorem, 16 double integral, 178 dual group, 193 duality theorem, 241 essential supremum (ess sup), 196 253 254 lode* nteaibd Stone-W eierstrasn Theo- rem, 2-14 extension theorem fTietxe), J1R factor group, 114 FatouV Lemma, )6 finite intersection property , S3 Fourier Transform on £,(-«>. »), Iff -> 19-20 on ti(G), 207 ff . 210 ff on £»(G), 237 Founer-Ptjpltjes Transform, 219 ff, 224 Fubim'a Theorem, 16, 179, 213 fundamental system of neighbor- hoods 74 Gel land theory, 43 (I topology 37 generalised mfpotent element, 57 sequence (convergence of) 116 Cauchy sequence 116 generated topology, 77 group algebra 27 dual, 193 factor, 114 general linear, 99 homogeneous 103 quotient, 114 regular, 106 topological, PS ununodular, 99 Haar covering function, 130 integral, 12$, 129, 172 measure, 126, 169, 172 Il&usdorff space, SO Hildw’i inequality, 17 homeomorphism, 62 homogeneous, 103 ideal, 4S maximal, 49 principal, W proper, 43 identity continuous partition of, 121 induced measure, 12S topology, 78 interior point, 71 inversion formulas L»(-«, »), 5 »),21 Li(G) 227 involution algebra with, 97 isomorphism, 123 iterated integral, 178 •Jordan Decomposition Theorem, 212 lattice linear vector, 159 Lcbesgue point, 5 limit point, 123 Liouville • Theorem 30 locally compact, 85 lower aemieontinuous functior 166 maximal ideal 49 measurable function, 12S mS IG1 set, 168 transformation 127-128 measure a!*olute continuity of, 243 complex Bore!, 211 outer, 64 product, 176 ff signed, 212 Index 255 Minkowski inequality, 17 modular function, 187 natural mapping, 110 neighborhood, 66 ff. fundamental system of, 74 symmetric, 104 topology, 66 ff. nilpotent element, 57 normal space, 117 normal subgroup, 110 normed algebra, 25 One-point Compactification, 118 open sets, 61 in terms of neighborhoods, 67 open mapping, 111 outer measure, 164 Parseval’s Theorem, 20, 244 Plancherel’s Theorem, 24, 235 Pontrjagin Duality Theorem, 241 positive definite function, 220 positive linear functional, 158 power set, 61 principal ideal, 50 product measures, 176 ff. product space, 78, 79 projection mapping, 84 proper ideal, 48 pth power summable, 1 quotient algebra, 50 group, 114 radical, 56 Radon-Nikodym Theorem, 245 regular_ content, 163 measure (inner, outer), 215 point, 37 topological space, 106 relative topology, 78 Representation Theorem, 160 Riesz, 245 resolvent operator, 41 Riemann-Lebesgue Lemma, 4 right Haar measure, 126, 172 ff. ring of sets, 125 semisimple, 57 separation axioms, 79 ff. o-algebra, 125 <r-bounded, 164 tr-ring, 125 simple function, 217 spectral radius, 42 spectrum of an element, 37 star (symmetric) algebra, 97 Stone-Weierstrass Theorem, 244 stronger topology, 77 subbasis (subbase), 75 subgroup, 109 closed, 110 normal, 114 subspace topology, 78 summable set, 168 support of a function, 107 symmetric neighborhood, 104 Tietze Extension Theorem, 118 ToDelli-Hobson Theorem, 17 topological divisor of zero, 53 topological group, 98 homogeneous, 103 locally compact, 107 normal subgroup, 114 regular, 106 subgroup of, 109 topological space, 61 ff. topology, 61 discrete, 61 Gel’fand, 87 generated, 77 induced, 78 relative (subspace), 78 trivial, 61 Tychonoff, 85 weak, 78 weaker (stronger), 77 256 Inden total variation, 211 Tychonoff Theorem, 85 topology, 85 weak topology, 78 weaker topology, 77 %\iener Theorem, CO i -section, 177 uniformly continuous, 107 ummodular group, 90 unit, 34 Urysohn’a Lemma, 118 y -«ection, 177 *ero set, 167 Zorn’s Lemma, 49